assempde

(Not recommended) Assemble finite element matrices and solve elliptic PDE

assempde is not recommended. Use solvepde instead.

Syntax

u = assempde(model,c,a,f)

u = assempde(b,p,e,t,c,a,f)

[Kc,Fc,B,ud] = assempde(___)

[Ks,Fs] = assempde(___)

[K,M,F,Q,G,H,R] = assempde(___)

[K,M,F,Q,G,H,R] = assempde(___,[],sdl)

u = assempde(K,M,F,Q,G,H,R)

[Ks,Fs] = assempde(K,M,F,Q,G,H,R)

[Kc,Fc,B,ud] = assempde(K,M,F,Q,G,H,R)

Description

u = assempde(model,c,a,f) solves the PDE

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

with geometry, boundary conditions, and finite element mesh in model, and coefficients c, a, and f. If the PDE is a system of equations (model.PDESystemSize > 1), then assempde solves the system of equations

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

u = assempde(b,p,e,t,c,a,f) solves the PDE with boundary conditions b, and finite element mesh (p,e,t).

[Kc,Fc,B,ud] = assempde(___), for any of the previous input syntaxes, assembles finite element matrices using the reduced linear system form, which eliminates any Dirichlet boundary conditions from the system of linear equations. You can calculate the solution u at node points by the command u = B*(Kc\Fc) + ud. See Reduced Linear System.

[Ks,Fs] = assempde(___) assembles finite element matrices that represent any Dirichlet boundary conditions using a stiff-spring approximation. You can calculate the solution u at node points by the command u = Ks\Fs. See Stiff-Spring Approximation.

[K,M,F,Q,G,H,R] = assempde(___) assembles finite element matrices that represent the PDE problem. This syntax returns all the matrices involved in converting the problem to finite element form. See Algorithms.

[K,M,F,Q,G,H,R] = assempde(___,[],sdl) restricts the finite element matrices to those that include the subdomain specified by the subdomain labels in sdl. The empty argument is required in this syntax for historic and compatibility reasons.

u = assempde(K,M,F,Q,G,H,R) returns the solution u based on the full collection of finite element matrices.

[Ks,Fs] = assempde(K,M,F,Q,G,H,R) returns finite element matrices that approximate Dirichlet boundary conditions using the stiff-spring approximation. See Algorithms.

[Kc,Fc,B,ud] = assempde(K,M,F,Q,G,H,R) returns finite element matrices that eliminate any Dirichlet boundary conditions from the system of linear equations. See Algorithms.

Examples

Solve a Scalar PDE

Solve an elliptic PDE on an L-shaped region.

Create a scalar PDE model. Incorporate the geometry of an L-shaped region.

model = createpde;
geometryFromEdges(model,@lshapeg);

Apply zero Dirichlet boundary conditions to all edges.

applyBoundaryCondition(model,Edge=1:model.Geometry.NumEdges,u=0);

Generate a finite element mesh.

generateMesh(model,GeometricOrder="linear");

Solve the PDE $$-\nabla \cdot (c\nabla u)+au=f$$ with parameters c = 1, a = 0, and f = 5.

c = 1;
a = 0;
f = 5;
u = assempde(model,c,a,f);

Plot the solution.

pdeplot(model,XYData=u)

3-D Elliptic Problem

Solve a 3-D elliptic PDE using a PDE model.

Create a PDE model container, import a 3-D geometry description, and view the geometry.

model = createpde;
importGeometry(model,'Block.stl');
pdegplot(model,FaceAlpha=0.5, ...
               FaceLabels="on")

Set zero Dirichlet conditions on faces 1 through 4 (the edges). Set Neumann conditions with g = -1 on face 6 and g = 1 on face 5.

applyBoundaryCondition(model,Face=1:4, ...
                             u=0);
applyBoundaryCondition(model,Face=6, ...
                             g=-1);
applyBoundaryCondition(model,Face=5, ...
                             g=1);

Set coefficients c = 1, a = 0, and f = 0.1.

c = 1;
a = 0;
f = 0.1;

Create a mesh and solve the problem.

msh = generateMesh(model);
u = assempde(model,c,a,f);

Plot the solution on the surface.

pdeplot3D(msh,ColorMapData=u)

2-D PDE Using [p,e,t] Mesh

Solve a 2-D PDE using the older syntax for mesh.

Create a circle geometry.

g = @circleg;

Set zero Dirichlet boundary conditions.

b = @circleb1;

Create a mesh for the geometry.

[p,e,t] = initmesh(g);

Solve the PDE $$-\nabla \cdot (c\nabla u)+au=f$$ with parameters c = 1, a = 0, and f = sin(x).

c = 1;
a = 0;
f = 'sin(x)';
u = assempde(b,p,e,t,c,a,f);

Plot the solution.

pdeplot(p,e,t,XYData=u)

Finite Element Matrices

Obtain the finite-element matrices that represent the problem using a reduced linear algebra representation of Dirichlet boundary conditions.

Create a scalar PDE model. Import a simple 3-D geometry.

model = createpde;
importGeometry(model,"Block.stl");

Set zero Dirichlet boundary conditions on all the geometry faces.

applyBoundaryCondition(model,"dirichlet", ...
                             Face=1:model.Geometry.NumFaces, ...
                             u=0);

Generate a mesh for the geometry.

generateMesh(model);

Obtain finite element matrices K, F, B, and ud that represent the equation $$-\nabla \cdot (c\nabla u)+au=f$$ with parameters $$c=1$$, $$a=0$$, and $$f=\log (1+x+\frac{y}{1+z})$$.

c = 1;
a = 0;
f = 'log(1+x+y./(1+z))';
[K,F,B,ud] = assempde(model,c,a,f);

You can obtain the solution u of the PDE at mesh nodes by executing the command

u = B*(K\F) + ud;

Generally, this solution is slightly more accurate than the stiff-spring solution, as calculated in the next example.

Stiff-Spring Finite Element Solution

Obtain the stiff-spring approximation of finite element matrices.

Create a scalar PDE model. Import a simple 3-D geometry.

model = createpde;
importGeometry(model,"Block.stl");

Set zero Dirichlet boundary conditions on all the geometry faces.

applyBoundaryCondition(model,Face=1:model.Geometry.NumFaces,u=0);

Generate a mesh for the geometry.

generateMesh(model);

Obtain finite element matrices Ks and Fs that represent the equation $$-\nabla \cdot (c\nabla u)+au=f$$ with parameters $$c=1$$, $$a=0$$, and $$f=\log (1+x+\frac{y}{1+z})$$.

c = 1;
a = 0;
f = "log(1+x+y./(1+z))";
[Ks,Fs] = assempde(model,c,a,f);

You can obtain the solution u of the PDE at mesh nodes by executing the command

u = Ks\Fs;

Generally, this solution is slightly less accurate than the reduced linear algebra solution, as calculated in the previous example.

Full Collection of Finite Element Matrices

Obtain the full collection of finite element matrices for an elliptic problem.

Import geometry and set up an elliptic problem with Dirichlet boundary conditions. The Torus.stl geometry has only one face, so you need set only one boundary condition.

model = createpde();
importGeometry(model,"Torus.stl");
applyBoundaryCondition(model,Face=1,u=0);
c = 1;
a = 0;
f = 1;
generateMesh(model);

Create the finite element matrices that represent this problem.

[K,M,F,Q,G,H,R] = ...
assempde(model,c,a,f);

Most of the resulting matrices are quite sparse. G, M, Q, and R are all zero sparse matrices.

howsparse = @(x)nnz(x)/numel(x);
disp(['Maximum fraction of nonzero' ...
      ' entries in K or H is ',...
      num2str(max(howsparse(K),howsparse(H)))])

Maximum fraction of nonzero entries in K or H is 0.0019281

To find the solution to the PDE, call assempde again.

u = assempde(K,M,F,Q,G,H,R);

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

b — Boundary conditions

boundary matrix | boundary file

Boundary conditions, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a file name. A boundary matrix is generally an export from the PDE Modeler app.

Example: b = 'circleb1', b = "circleb1", or b = @circleb1

Data Types: double | char | string | function_handle

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

e — Mesh edges

matrix

Mesh edges, specified as a 7-by-Ne matrix of edges, where Ne is the number of edges 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

K — Stiffness matrix

sparse matrix | full matrix

Stiffness matrix, specified as a sparse matrix or full matrix. Generally, you obtain K from a previous call to assema or assempde. For the meaning of stiffness matrix, see Elliptic Equations.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

M — Mass matrix

sparse matrix | full matrix

Mass matrix, specified as a sparse matrix or full matrix. Generally, you obtain M from a previous call to assema or assempde. For the meaning of mass matrix, see Elliptic Equations.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

F — Finite element f representation

vector

Finite element f representation, specified as a vector. Generally, you obtain F from a previous call to assema or assempde. For the meaning of this representation, see Elliptic Equations.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

Q — Neumann boundary condition matrix

sparse matrix | full matrix

Neumann boundary condition matrix, specified as a sparse matrix or full matrix. Generally, you obtain Q from a previous call to assemb or assempde. For the meaning of this matrix, see Elliptic Equations.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

G — Neumann boundary condition vector

sparse vector | full vector

Neumann boundary condition vector, specified as a sparse vector or full vector. Generally, you obtain G from a previous call to assemb or assempde. For the meaning of this vector, see Elliptic Equations.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

H — Dirichlet boundary condition matrix

sparse matrix | full matrix

Dirichlet boundary condition matrix, specified as a sparse matrix or full matrix. Generally, you obtain H from a previous call to assemb or assempde. For the meaning of this matrix, see Algorithms.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

R — Dirichlet boundary condition vector

sparse vector | full vector

Dirichlet boundary condition vector, specified as a sparse vector or full vector. Generally, you obtain R from a previous call to assemb or assempde. For the meaning of this vector, see Algorithms.

Example: [K,M,F,Q,G,H,R] = assempde(model,c,a,f)

Data Types: double
Complex Number Support: Yes

sdl — Subdomain labels

vector of positive integers

Subdomain labels, specified as a vector of positive integers. For 2-D geometry only. View the subdomain labels in your geometry using the command

pdegplot(g,'SubdomainLabels','on')

Example: sdl = [1,3:5];

Data Types: double

Output Arguments

u — PDE solution

vector

PDE solution, returned as a vector.

• If the PDE is scalar, meaning only one equation, then u is a column vector representing the solution u at each node in the mesh. u(i) is the solution at the ith column of model.Mesh.Nodes or the ith column of p.

• If the PDE is a system of N > 1 equations, then u is a column vector with N*Np elements, where Np is the number of nodes in the mesh. The first Np elements of u represent the solution of equation 1, then next Np elements represent the solution of equation 2, etc.

To obtain the solution at an arbitrary point in the geometry, use pdeInterpolant.

Kc — Stiffness matrix

sparse matrix

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

u1 = Kc\Fc returns the solution on the non-Dirichlet points. To obtain the solution u at the nodes of the mesh,

u = B*(Kc\Fc) + ud

Generally, Kc, Fc, B, and ud make a slower but more accurate solution than Ks and Fs.

Fc — Load vector

vector

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

u = B*(Kc\Fc) + ud

Generally, Kc, Fc, B, and ud make a slower but more accurate solution than Ks and Fs.

B — Dirichlet nullspace

sparse matrix

Dirichlet nullspace, returned as a sparse matrix. See Algorithms.

u = B*(Kc\Fc) + ud

Generally, Kc, Fc, B, and ud make a slower but more accurate solution than Ks and Fs.

ud — Dirichlet vector

vector

Dirichlet vector, returned as a vector. See Algorithms.

u = B*(Kc\Fc) + ud

Generally, Kc, Fc, B, and ud make a slower but more accurate solution than Ks and Fs.

Ks — Stiffness matrix corresponding to the stiff-spring approximation for Dirichlet boundary condition

sparse matrix

Finite element matrix for stiff-spring approximation, returned as a sparse matrix. See Algorithms.

To obtain the solution u at the nodes of the mesh,

u = Ks\Fs.

Generally, Ks and Fs make a quicker but less accurate solution than Kc, Fc, B, and ud.

Fs — Load vector corresponding to the stiff-spring approximation for Dirichlet boundary condition

vector

Load vector corresponding to the stiff-spring approximation for Dirichlet boundary condition, returned as a vector. See Algorithms.

To obtain the solution u at the nodes of the mesh,

u = Ks\Fs.

Generally, Ks and Fs make a quicker but less accurate solution than Kc, Fc, B, and ud.

K — Stiffness matrix

sparse matrix

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

K represents the stiffness matrix alone, unlike Kc or Ks, which are stiffness matrices combined with other terms to enable immediate solution of a PDE.

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

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.

F represents the load vector alone, unlike Fc or Fs, which are load vectors combined with other terms to enable immediate solution of a PDE.

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

Q — Neumann boundary condition matrix

sparse matrix

Neumann boundary condition matrix, returned as a sparse matrix. See Elliptic Equations.

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

G — Neumann boundary condition vector

sparse vector

Neumann boundary condition vector, returned as a sparse vector. See Elliptic Equations.

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

H — Dirichlet matrix

sparse matrix

Dirichlet matrix, returned as a sparse matrix. See Algorithms.

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

R — Dirichlet vector

sparse vector

Dirichlet vector, returned as a sparse vector. See Algorithms.

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

More About

Reduced Linear System

This form of the finite element matrices eliminates Dirichlet conditions from the problem using a linear algebra approach. The finite element matrices reduce to the solution u = B*(Kc\Fc) + ud, where B spans the null space of the columns of H (the Dirichlet condition matrix representing hu = r). R is the Dirichlet condition vector for Hu = R. ud is the vector of boundary condition solutions for the Dirichlet conditions. u1 = Kc\Fc returns the solution on the non-Dirichlet points.

See Systems of PDEs for details on the approach used to eliminate Dirichlet conditions.

Stiff-Spring Approximation

This form of the finite element matrices converts Dirichlet boundary conditions to Neumann boundary conditions using a stiff-spring approximation. Using this approximation, assempde returns a matrix Ks and a vector Fs that represent the combined finite element matrices. The approximate solution u is u = Ks\Fs.

See Elliptic Equations. For details of the stiff-spring approximation, see Systems of PDEs.

Algorithms

Elliptic Equations

Partial Differential Equation Toolbox™ solves equations of the form

$$m\frac{{\partial }^{2}u}{\partial t^2}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\nabla u\right)+au=f$$

When the m and d coefficients are 0, this reduces to

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

which the documentation calls an elliptic equation, whether or not the equation is elliptic in the mathematical sense. The equation holds in Ω, where Ω is a bounded domain in two or three dimensions. c, a, f, and the unknown solution u are complex functions defined on Ω. c can also be a 2-by-2 matrix function on Ω. The boundary conditions specify a combination of u and its normal derivative on the boundary:

• Dirichlet: hu = r on the boundary ∂Ω.

• Generalized Neumann: $$\overrightarrow{n}$$ · (c∇u) + qu = g on ∂Ω.

• Mixed: Only applicable to systems. A combination of Dirichlet and generalized Neumann.

$$\overrightarrow{n}$$ is the outward unit normal. g, q, h, and r are functions defined on ∂Ω.

Our nomenclature deviates slightly from the tradition for potential theory, where a Neumann condition usually refers to the case q = 0 and our Neumann would be called a mixed condition. In some contexts, the generalized Neumann boundary conditions is also referred to as the Robin boundary conditions. In variational calculus, Dirichlet conditions are also called essential boundary conditions and restrict the trial space. Neumann conditions are also called natural conditions and arise as necessary conditions for a solution. The variational form of the Partial Differential Equation Toolbox equation with Neumann conditions is given below.

The approximate solution to the elliptic PDE is found in three steps:

1. Describe the geometry of the domain Ω and the boundary conditions. For 2-D geometry, create geometry using the PDE Modeler app or through MATLAB^® files. For 3-D geometry, import the geometry in STL file format.

2. Build a triangular mesh on the domain Ω. The software has mesh generating and mesh refining facilities. A mesh is described by three matrices of fixed format that contain information about the mesh points, the boundary segments, and the elements.

3. Discretize the PDE and the boundary conditions to obtain a linear system Ku = F. The unknown vector u contains the values of the approximate solution at the mesh points, the matrix K is assembled from the coefficients c, a, h, and q and the right-hand side F contains, essentially, averages of f around each mesh point and contributions from g. Once the matrices K and F are assembled, you have the entire MATLAB environment at your disposal to solve the linear system and further process the solution.

More elaborate applications make use of the Finite Element Method (FEM) specific information returned by the different functions of the software. Therefore we quickly summarize the theory and technique of FEM solvers to enable advanced applications to make full use of the computed quantities.

FEM can be summarized in the following sentence: Project the weak form of the differential equation onto a finite-dimensional function space. The rest of this section deals with explaining the preceding statement.

We start with the weak form of the differential equation. Without restricting the generality, we assume generalized Neumann conditions on the whole boundary, since Dirichlet conditions can be approximated by generalized Neumann conditions. In the simple case of a unit matrix h, setting g = qr and then letting q → ∞ yields the Dirichlet condition because division with a very large q cancels the normal derivative terms. The actual implementation is different, since the preceding procedure may create conditioning problems. The mixed boundary condition of the system case requires a more complicated treatment, described in Systems of PDEs.

Assume that u is a solution of the differential equation. Multiply the equation with an arbitrary test function v and integrate on Ω:

$${\displaystyle \underset{\Omega }{\int }\left(-\left(\nabla ·c\nabla u\right)v+auv\right)dx}={\displaystyle \underset{\Omega }{\int }fvdx}$$

Integrate by parts (i.e., use Green's formula) to obtain

$${\displaystyle \underset{\Omega }{\int }\left(\left(c\nabla u\right)·\nabla v+auv\right)dx}-{\displaystyle \underset{\partial \Omega }{\int }\overrightarrow{n}·\left(c\nabla u\right)vds}={\displaystyle \underset{\Omega }{\int }fvdx}$$

The boundary integral can be replaced by the boundary condition:

$${\displaystyle \underset{\Omega }{\int }\left(\left(c\nabla u\right)·\nabla v+auv\right)dx}-{\displaystyle \underset{\partial \Omega }{\int }\left(-qu+g\right)vds}={\displaystyle \underset{\Omega }{\int }fvdx}$$

Replace the original problem with Find u such that

$${\displaystyle \underset{\Omega }{\int }\left(\left(c\nabla u\right)·\nabla v+auv-fv\right)dx}-{\displaystyle \underset{\partial \Omega }{\int }\left(-qu+g\right)vds}=0\forall v$$

This equation is called the variational, or weak, form of the differential equation. Obviously, any solution of the differential equation is also a solution of the variational problem. The reverse is true under some restrictions on the domain and on the coefficient functions. The solution of the variational problem is also called the weak solution of the differential equation.

The solution u and the test functions v belong to some function space V. The next step is to choose an Np-dimensional subspace $$V_{N_p}\subset V$$. Project the weak form of the differential equation onto a finite-dimensional function space simply means requesting u and v to lie in $$V_{N_p}$$ rather than V. The solution of the finite dimensional problem turns out to be the element of $$V_{N_p}$$ that lies closest to the weak solution when measured in the energy norm. Convergence is guaranteed if the space $$V_{N_p}$$ tends to V as N_p→∞. Since the differential operator is linear, we demand that the variational equation is satisfied for N_p test-functions Φ_i ∊$$V_{N_p}$$ that form a basis, i.e.,

$${\displaystyle \underset{\Omega }{\int }\left(\left(c\nabla u\right)·\nabla {\varphi }_i+au{\varphi }_i-f{\varphi }_i\right)dx}-{\displaystyle \underset{\partial \Omega }{\int }\left(-qu+g\right){\varphi }_{i}ds}=0,i=1,\mathrm{...},N_p$$

Expand u in the same basis of $$V_{N_p}$$ elements

$$u(x)={\displaystyle \sum _{j=1}^{N_p}{U}_j{\varphi }_j(x)}$$

and obtain the system of equations

$${\displaystyle \sum _{j=1}^{N_p}\left({\displaystyle \underset{\Omega }{\int }\left(\left(c\nabla {\varphi }_j\right)·\nabla {\varphi }_i+a{\varphi }_j{\varphi }_i\right)dx}+{\displaystyle \underset{\partial \Omega }{\int }q{\varphi }_j{\varphi }_{i}ds}\right)}{U}_j={\displaystyle \underset{\Omega }{\int }f{\varphi }_{i}dx}+{\displaystyle \underset{\partial \Omega }{\int }g{\varphi }_{i}ds},i=1,\mathrm{...},N_p$$

Use the following notations:

$$K_{i,j}={\displaystyle \underset{\Omega }{\int }\left(c\nabla {\varphi }_j\right)\cdot \nabla {\varphi }_{i}dx}\text{(stiffness matrix)}$$

$$M_{i,j}={\displaystyle \underset{\Omega }{\int }a{\varphi }_j{\varphi }_{i}dx}\text{(mass matrix)}$$

$$Q_{i,j}={\displaystyle \underset{\partial \Omega }{\int }q{\varphi }_j{\varphi }_{i}ds}$$

$$F_i={\displaystyle \underset{\Omega }{\int }f{\varphi }_{i}dx}$$

$$G_i={\displaystyle \underset{\partial \Omega }{\int }g{\varphi }_{i}ds}$$

and rewrite the system in the form

(K + M + Q)U = F + G.

(1)

K, M, and Q are N_p-by-N_p matrices, and F and G are N_p-vectors. K, M, and F are produced by assema, while Q, G are produced by assemb. When it is not necessary to distinguish K, M, and Q or F and G, we collapse the notations to KU = F, which form the output of assempde.

When the problem is self-adjoint and elliptic in the usual mathematical sense, the matrix K + M + Q becomes symmetric and positive definite. Many common problems have these characteristics, most notably those that can also be formulated as minimization problems. For the case of a scalar equation, K, M, and Q are obviously symmetric. If c(x) ≥ δ > 0, a(x) ≥ 0 and q(x) ≥ 0 with q(x) > 0 on some part of ∂Ω, then, if U ≠ 0.

$$U^T\left(K+M+Q\right)U={\displaystyle \underset{\Omega }{\int }\left(c{\left|u\right|}^2+a{u}^2\right)dx}+{\displaystyle \underset{\partial \Omega }{\int }q{u}^{2}ds}>0,\text{if }U\ne 0$$

U^T(K + M + Q)U is the energy norm. There are many choices of the test-function spaces. The software uses continuous functions that are linear on each element of a 2-D mesh, and are linear or quadratic on elements of a 3-D mesh. Piecewise linearity guarantees that the integrals defining the stiffness matrix K exist. Projection onto $$V_{N_p}$$ is nothing more than linear interpolation, and the evaluation of the solution inside an element is done just in terms of the nodal values. If the mesh is uniformly refined, $$V_{N_p}$$ approximates the set of smooth functions on Ω.

A suitable basis for $$V_{N_p}$$ in 2-D is the set of “tent” or “hat” functions ϕ_i. These are linear on each element and take the value 0 at all nodes x_j except for x_i. For the definition of basis functions for 3-D geometry, see Finite Element Basis for 3-D. Requesting ϕ_i(x_i) = 1 yields the very pleasant property

$$u\left(x_i\right)={\displaystyle \sum _{j=1}^{N_p}{U}_j{\varphi }_j\left(x_i\right)}=U_i$$

That is, by solving the FEM system we obtain the nodal values of the approximate solution. The basis function ϕ_i vanishes on all the elements that do not contain the node x_i. The immediate consequence is that the integrals appearing in K_i,j, M_i,j, Q_i,j, F_i and G_i only need to be computed on the elements that contain the node x_i. Secondly, it means that K_i,j andM_i,j are zero unless x_i and x_j are vertices of the same element and thus K and M are very sparse matrices. Their sparse structure depends on the ordering of the indices of the mesh points.

The integrals in the FEM matrices are computed by adding the contributions from each element to the corresponding entries (i.e., only if the corresponding mesh point is a vertex of the element). This process is commonly called assembling, hence the name of the function assempde.

The assembling routines scan the elements of the mesh. For each element they compute the so-called local matrices and add their components to the correct positions in the sparse matrices or vectors.

The discussion now specializes to triangular meshes in 2-D. The local 3-by-3 matrices contain the integrals evaluated only on the current triangle. The coefficients are assumed constant on the triangle and they are evaluated only in the triangle barycenter. The integrals are computed using the midpoint rule. This approximation is optimal since it has the same order of accuracy as the piecewise linear interpolation.

Consider a triangle given by the nodes P₁, P₂, and P₃ as in the following figure.

Note

The local 3-by-3 matrices contain the integrals evaluated only on the current triangle. The coefficients are assumed constant on the triangle and they are evaluated only in the triangle barycenter.

The simplest computations are for the local mass matrix m:

$$m_{i,j}={\displaystyle \underset{\Delta P_1{P}_2{P}_3}{\int }a\left(P_c\right){\varphi }_i\left(x\right){\varphi }_j\left(x\right)dx}=a\left(P_c\right)\frac{\text{area}\left(\Delta P_1{P}_2{P}_3\right)}{12}\left(1+{\delta }_{i,j}\right)$$

where P_c is the center of mass of Δ P₁P₂P₃, i.e.,

$$P_c=\frac{P_1+P_2+P_3}{3}$$

The contribution to the right side F is just

$$f_i=f\left(P_c\right)\frac{\text{area}\left(\Delta P_1{P}_2{P}_3\right)}{3}$$

For the local stiffness matrix we have to evaluate the gradients of the basis functions that do not vanish on P₁P₂P₃. Since the basis functions are linear on the triangle P₁P₂P₃, the gradients are constants. Denote the basis functions ϕ₁, ϕ₂, and ϕ₃ such that ϕ(P_i) = 1. If P₂ – P₃ = [x₁,y₁]^T then we have that

$$\nabla {\varphi }_1=\frac{1}{2\text{area}\left(\Delta P_1{P}_2{P}_3\right)}\left[\begin{array}{c}{y}_1\\ -x_1\end{array}\right]$$

and after integration (taking c as a constant matrix on the triangle)

$$k_{i,j}=\frac{1}{4\text{area}\left(\Delta P_1{P}_2{P}_3\right)}\left[y_j,-x_j\right]c\left(P_c\right)\left[\begin{array}{c}{y}_1\\ -x_1\end{array}\right]$$

If two vertices of the triangle lie on the boundary ∂Ω, they contribute to the line integrals associated to the boundary conditions. If the two boundary points are P₁ and P₂, then we have

$$Q_{i,j}=q\left(P_b\right)\frac{\Vert P_1-P_2\Vert }{6}\left(1+{\delta }_{i,j}\right),i,j=1,2$$

and

$$G_i=g\left(P_b\right)\frac{\Vert P_1-P_2\Vert }{2},i=1,2$$

where P_b is the midpoint of P₁P₂.

For each triangle the vertices P_m of the local triangle correspond to the indices i_m of the mesh points. The contributions of the individual triangle are added to the matrices such that, e.g.,

$$K_{i_m,i_n}t\leftarrow K_{i_m,i_n}+k_{m,n},m,n=1,2,3$$

This is done by the function assempde. The gradients and the areas of the triangles are computed by the function pdetrg.

The Dirichlet boundary conditions are treated in a slightly different manner. They are eliminated from the linear system by a procedure that yields a symmetric, reduced system. The function assempde can return matrices K, F, B, and ud such that the solution is u = Bv + ud where Kv = F. u is an N_p-vector, and if the rank of the Dirichlet conditions is rD, then v has N_p – rD components.

To summarize, assempde performs the following steps to obtain a solution u to an elliptic PDE:

1. Generate the finite element matrices [K,M,F,Q,G,H,R]. This step is equivalent to calling assema to generate the matrices K, M, and F, and also calling assemb to generate the matrices Q, G, H, and R.

2. Generate the combined finite element matrices [Kc,Fc,B,ud]. 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.

3. Calculate the solution u via

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

assempde uses one of two algorithms for assembling a problem into combined finite element matrix form. A reduced linear system form leads to immediate solution via linear algebra. You choose the algorithm by the number of outputs. For the reduced linear system form, request four outputs:

[Kc,Fc,B,ud] = assempde(_)

For the stiff-spring approximation, request two outputs:

[Ks,Fs] = assempde(_)

For details, see Reduced Linear System and Stiff-Spring Approximation.

Systems of PDEs

Partial Differential Equation Toolbox software can also handle systems of N partial differential equations over the domain Ω. We have the elliptic system

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

the parabolic system

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

the hyperbolic system

$$d\frac{{\partial }^{2}u}{\partial t^2}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

and the eigenvalue system

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

where c is an N-by-N-by-D-by-D tensor, and D is the geometry dimensions, 2 or 3.

For 2-D systems, the notation $$\nabla \cdot (c\otimes \nabla u)$$ represents an N-by-1 matrix with an (i,1)-component

$${\displaystyle \sum _{j=1}^N\left(\frac{\partial }{\partial x}{c}_{i,j,1,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial x}{c}_{i,j,1,2}\frac{\partial }{\partial y}+\frac{\partial }{\partial y}{c}_{i,j,2,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial y}{c}_{i,j,2,2}\frac{\partial }{\partial y}\right)}{u}_j$$

For 3-D systems, the notation $$\nabla \cdot (c\otimes \nabla u)$$ represents an N-by-1 matrix with an (i,1)-component

$$\begin{array}{l}{\displaystyle \sum _{j=1}^N\left(\frac{\partial }{\partial x}{c}_{i,j,1,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial x}{c}_{i,j,1,2}\frac{\partial }{\partial y}+\frac{\partial }{\partial x}{c}_{i,j,1,3}\frac{\partial }{\partial z}\right)u_j}\\ +{\displaystyle \sum _{j=1}^N\left(\frac{\partial }{\partial y}{c}_{i,j,2,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial y}{c}_{i,j,2,2}\frac{\partial }{\partial y}+\frac{\partial }{\partial y}{c}_{i,j,2,3}\frac{\partial }{\partial z}\right)u_j}\\ +{\displaystyle \sum _{j=1}^N\left(\frac{\partial }{\partial z}{c}_{i,j,3,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial z}{c}_{i,j,3,2}\frac{\partial }{\partial y}+\frac{\partial }{\partial z}{c}_{i,j,3,3}\frac{\partial }{\partial z}\right)u_j}\end{array}$$

The symbols a and d denote N-by-N matrices, and f denotes a column vector of length N.

The elements c_ijkl, a_ij, d_ij, and f_i of c, a, d, and f are stored row-wise in the MATLAB matrices c, a, d, and f. The case of identity, diagonal, and symmetric matrices are handled as special cases. For the tensor c_ijkl this applies both to the indices i and j, and to the indices k and l.

Partial Differential Equation Toolbox software does not check the ellipticity of the problem, and it is quite possible to define a system that is not elliptic in the mathematical sense. The preceding procedure that describes the scalar case is applied to each component of the system, yielding a symmetric positive definite system of equations whenever the differential system possesses these characteristics.

The boundary conditions now in general are mixed, i.e., for each point on the boundary a combination of Dirichlet and generalized Neumann conditions,

$$\begin{array}{l}hu=r\\ n·\left(c\otimes \nabla u\right)+qu=g+h^{\prime }\mu \end{array}$$

For 2-D systems, the notation $$n·\left(c\otimes \nabla u\right)$$ represents an N-by-1 matrix with (i,1)-component

$${\displaystyle \sum _{j=1}^N\left(\cos (\alpha )c_{i,j,1,1}\frac{\partial }{\partial x}+\cos (\alpha )c_{i,j,1,2}\frac{\partial }{\partial y}+\sin (\alpha )c_{i,j,2,1}\frac{\partial }{\partial x}+\sin (\alpha )c_{i,j,2,2}\frac{\partial }{\partial y}\right)}{u}_j$$

where the outward normal vector of the boundary is $$n=\left(\cos (\alpha ),\sin (\alpha )\right)$$.

For 3-D systems, the notation $$n·\left(c\otimes \nabla u\right)$$ represents an N-by-1 matrix with (i,1)-component

$$\begin{array}{l}{\displaystyle \sum _{j=1}^N\left(\cos (\alpha )c_{i,j,1,1}\frac{\partial }{\partial x}+\cos (\alpha )c_{i,j,1,2}\frac{\partial }{\partial y}+\cos (\alpha )c_{i,j,1,3}\frac{\partial }{\partial z}\right)}{u}_j\\ +{\displaystyle \sum _{j=1}^N\left(\cos (\beta )c_{i,j,2,1}\frac{\partial }{\partial x}+\cos (\beta )c_{i,j,2,2}\frac{\partial }{\partial y}+\cos (\beta )c_{i,j,2,3}\frac{\partial }{\partial z}\right)}{u}_j\\ +{\displaystyle \sum _{j=1}^N\left(\cos (\gamma )c_{i,j,3,1}\frac{\partial }{\partial x}+\cos (\gamma )c_{i,j,3,2}\frac{\partial }{\partial y}+\cos (\gamma )c_{i,j,3,3}\frac{\partial }{\partial z}\right)}{u}_j\end{array}$$

where the outward normal to the boundary is

$$n=\left(\cos \left(\alpha \right),\cos \left(\beta \right),\cos \left(\gamma \right)\right)$$

There are M Dirichlet conditions and the h-matrix is M-by-N, M ≥ 0. The generalized Neumann condition contains a source $$h^{\prime }\mu$$, where the Lagrange multipliers μ are computed such that the Dirichlet conditions become satisfied. In a structural mechanics problem, this term is exactly the reaction force necessary to satisfy the kinematic constraints described by the Dirichlet conditions.

The rest of this section details the treatment of the Dirichlet conditions and may be skipped on a first reading.

Partial Differential Equation Toolbox software supports two implementations of Dirichlet conditions. The simplest is the “Stiff Spring” model, so named for its interpretation in solid mechanics. See Elliptic Equations for the scalar case, which is equivalent to a diagonal h-matrix. For the general case, Dirichlet conditions

hu = r

are approximated by adding a term

$$L(h^{\prime }hu-h^{\prime }r)$$

to the equations KU = F, where L is a large number such as 10⁴ times a representative size of the elements of K.

When this number is increased, hu = r will be more accurately satisfied, but the potential ill-conditioning of the modified equations will become more serious.

The second method is also applicable to general mixed conditions with nondiagonal h, and is free of the ill-conditioning, but is more involved computationally. Assume that there are N_p nodes in the mesh. Then the number of unknowns is N_pN = N_u. When Dirichlet boundary conditions fix some of the unknowns, the linear system can be correspondingly reduced. This is easily done by removing rows and columns when u values are given, but here we must treat the case when some linear combinations of the components of u are given, hu = r. These are collected into HU = R where H is an M-by-N_u matrix and R is an M-vector.

With the reaction force term the system becomes

KU +H´ µ = F

HU = R.

The constraints can be solved for M of the U-variables, the remaining called V, an N_u – M vector. The null space of H is spanned by the columns of B, and U = BV + u_d makes U satisfy the Dirichlet conditions. A permutation to block-diagonal form exploits the sparsity of H to speed up the following computation to find B in a numerically stable way. µ can be eliminated by pre-multiplying by B´ since, by the construction, HB = 0 or B´H´ = 0. The reduced system becomes

B´ KBV = B´ F – B´Ku_d

which is symmetric and positive definite if K is.

Finite Element Basis for 3-D

The finite element method for 3-D geometry is similar to the 2-D method described in Elliptic Equations. The main difference is that the elements in 3-D geometry are tetrahedra, which means that the basis functions are different from those in 2-D geometry.

It is convenient to map a tetrahedron to a canonical tetrahedron with a local coordinate system (r,s,t).

In local coordinates, the point p1 is at (0,0,0), p2 is at (1,0,0), p3 is at (0,1,0), and p4 is at (0,0,1).

For a linear tetrahedron, the basis functions are

$$\begin{array}{c}{\varphi }_1=1-r-s-t\\ {\varphi }_2=r\\ {\varphi }_3=s\\ {\varphi }_4=t\end{array}$$

For a quadratic tetrahedron, there are additional nodes at the edge midpoints.

The corresponding basis functions are

$$\begin{array}{c}{\varphi }_1=2{\left(1-r-s-t\right)}^2-\left(1-r-s-t\right)\\ {\varphi }_2=2r^2-r\\ {\varphi }_3=2s^2-s\\ {\varphi }_4=2t^2-t\\ {\varphi }_5=4r\left(1-r-s-t\right)\\ {\varphi }_6=4rs\\ {\varphi }_7=4s\left(1-r-s-t\right)\\ {\varphi }_8=4t\left(1-r-s-t\right)\\ {\varphi }_9=4rt\\ {\varphi }_{10}=4st\end{array}$$

As in the 2-D case, a 3-D basis function ϕ_i takes the value 0 at all nodes j, except for node i, where it takes the value 1.

Version History

Introduced before R2006a

R2016a: Not recommended

assempde is not recommended. Use solvepde instead. There are no plans to remove assempde.