specifyCoefficients

Specify coefficients in a PDE model

Syntax

specifyCoefficients(model,Name,Value)

specifyCoefficients(model,Name,Value,RegionType,RegionID)

CA = specifyCoefficients(___)

Description

Coefficients of a PDE

solvepde solves PDEs of the form

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

solvepdeeig solves PDE eigenvalue problems of the form

$$\begin{array}{l}-\nabla ·\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\nabla u\right)+au={\lambda }^{2}mu\end{array}$$

specifyCoefficients defines the coefficients m, d, c, a, and f in the PDE model.

specifyCoefficients(model,Name,Value) defines the specified coefficients in each Name to each associated Value, and includes them in model. You must specify all of these names: m, d, c, a, and f. This syntax applies coefficients to the entire geometry.

Note

Include geometry in model before using specifyCoefficients.

specifyCoefficients(model,Name,Value,RegionType,RegionID) assigns coefficients for a specified geometry region.

CA = specifyCoefficients(___) returns a handle to the coefficient assignment object in model.

Examples

Specify Poisson's Equation

Specify the coefficients for Poisson's equation $$-\nabla \cdot \nabla u=1$$.

solvepde addresses equations of the form

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

Therefore, the coefficients for Poisson's equation are $$m=0$$, $$d=0$$, $$c=1$$, $$a=0$$, $$f=1$$. Include these coefficients in a PDE model of the L-shaped membrane.

model = createpde();
geometryFromEdges(model,@lshapeg);
specifyCoefficients(model,"m",0,...
                          "d",0,...
                          "c",1,...
                          "a",0,...
                          "f",1);

Specify zero Dirichlet boundary conditions, mesh the model, and solve the PDE.

applyBoundaryCondition(model,"dirichlet", ...
                             "Edge",1:model.Geometry.NumEdges, ...
                             "u",0);
generateMesh(model,"Hmax",0.25);
results = solvepde(model);

View the solution.

pdeplot(model,"XYData",results.NodalSolution)

Coefficient Handle for Nonconstant Coefficients

Specify coefficients for Poisson's equation in 3-D with a nonconstant source term, and obtain the coefficient object.

The equation coefficients are $$m=0$$, $$d=0$$, $$c=1$$, $$a=0$$. For the nonconstant source term, take $$f=y^2\tanh \left(z\right)/1000$$.

f = @(location,state)location.y.^2.*tanh(location.z)/1000;

Set the coefficients in a 3-D rectangular block geometry.

model = createpde();
importGeometry(model,"Block.stl");
CA = specifyCoefficients(model,"m",0,...
                               "d",0,...
                               "c",1,...
                               "a",0,...
                               "f",f)

CA = 
  CoefficientAssignment with properties:

    RegionType: 'cell'
      RegionID: 1
             m: 0
             d: 0
             c: 1
             a: 0
             f: @(location,state)location.y.^2.*tanh(location.z)/1000

Set zero Dirichlet conditions on face 1, mesh the geometry, and solve the PDE.

applyBoundaryCondition(model,"dirichlet","Face",1,"u",0);
generateMesh(model);
results = solvepde(model);

View the solution on the surface.

pdeplot3D(model,"ColorMapData",results.NodalSolution)

Specify Coefficients Depending On Subdomain

Create a scalar PDE model with the L-shaped membrane as the geometry. Plot the geometry and subdomain labels.

model = createpde();
geometryFromEdges(model,@lshapeg);
pdegplot(model,"FaceLabels","on")
axis equal
ylim([-1.1,1.1])

Set the c coefficient to 1 in all domains, but the f coefficient to 1 in subdomain 1, 5 in subdomain 2, and -8 in subdomain 3. Set all other coefficients to 0.

specifyCoefficients(model,"m",0,"d",0,"c",1,"a",0,"f",1,"Face",1);
specifyCoefficients(model,"m",0,"d",0,"c",1,"a",0,"f",5,"Face",2);
specifyCoefficients(model,"m",0,"d",0,"c",1,"a",0,"f",-8,"Face",3);

Set zero Dirichlet boundary conditions to all edges. Create a mesh, solve the PDE, and plot the result.

applyBoundaryCondition(model,"dirichlet", ...
                             "Edge",1:model.Geometry.NumEdges, ...
                             "u",0);
generateMesh(model,"Hmax",0.25);
results = solvepde(model);
pdeplot(model,"XYData",results.NodalSolution)

Input Arguments

model — PDE model

PDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde

Name-Value Arguments

Example: specifyCoefficients(model,"m",0,"d",0,"c",1,"a",0,"f",@fcoeff)

Note

You must specify all of these names: m, d, c, a, and f.

m — Second-order time derivative coefficient

scalar | column vector | function handle

Second-order time derivative coefficient, specified as a scalar, column vector, or function handle. For details on the sizes, and for details of the function handle form of the coefficient, see m, d, or a Coefficient for specifyCoefficients.

Specify 0 on the entire geometry if the term is not part of your problem. If this coefficient is zero in one subdomain of the geometry, it must be zero on the entire geometry.

Example: specifyCoefficients("m",@mcoef,"d",0,"c",1,"a",0,"f",1,"Face",1:4)

Data Types: double | function_handle
Complex Number Support: Yes

d — First-order time derivative coefficient

scalar | column vector | function handle

First-order time derivative coefficient, specified as a scalar, column vector, or function handle. For details on the sizes, and for details of the function handle form of the coefficient, see m, d, or a Coefficient for specifyCoefficients.

Note

If the m coefficient is nonzero, d must be 0 or a matrix, and not a function handle. See d Coefficient When m is Nonzero.

Specify 0 on the entire geometry if the term is not part of your problem. If this coefficient is zero in one subdomain of the geometry, it must be zero on the entire geometry.

Example: specifyCoefficients("m",0,"d",@dcoef,"c",1,"a",0,"f",1,"Face",1:4)

Data Types: double | function_handle
Complex Number Support: Yes

c — Second-order space derivative coefficient

scalar | column vector | function handle

Second-order space derivative coefficient, specified as a scalar, column vector, or function handle. For details on the sizes, and for details of the function handle form of the coefficient, see c Coefficient for specifyCoefficients.

This coefficient must not be zero in one subdomain of the geometry while nonzero in another subdomain.

Example: specifyCoefficients("m",0,"d",0,"c",@ccoef,"a",0,"f",1,"Face",1:4)

Data Types: double | function_handle
Complex Number Support: Yes

a — Solution multiplier coefficient

scalar | column vector | function handle

Solution multiplier coefficient, specified as a scalar, column vector, or function handle. For details on the sizes, and for details of the function handle form of the coefficient, see m, d, or a Coefficient for specifyCoefficients.

Specify 0 if the term is not part of your problem. This coefficient can be zero in one subdomain and nonzero in another.

Example: specifyCoefficients("m",0,"d",0,"c",1,"a",@acoef,"f",1,"Face",1:4)

Data Types: double | function_handle
Complex Number Support: Yes

f — Source coefficient

scalar | column vector | function handle

Source coefficient, specified as a scalar, column vector, or function handle. For details on the sizes, and for details of the function handle form of the coefficient, see f Coefficient for specifyCoefficients.

Specify 0 if the term is not part of your problem. This coefficient can be zero in one subdomain and nonzero in another.

Example: specifyCoefficients("m",0,"d",0,"c",1,"a",0,"f",@fcoeff,"Face",1:4)

Data Types: double | function_handle
Complex Number Support: Yes

RegionType — Geometric region type

"Face" for a 2-D model | "Cell" for a 3-D model

Geometric region type, specified as "Face" or "Cell".

Example: specifyCoefficients("m",0,"d",0,"c",1,"a",0,"f",10,"Cell",2)

Data Types: char | string

RegionID — Geometric region ID

vector of positive integers

Geometric region ID, specified as a vector of positive integers. Find the region IDs by using pdegplot.

Example: specifyCoefficients("m",0,"d",0,"c",1,"a",0,"f",10,"Cell",1:3)

Data Types: double

Output Arguments

CA — Coefficient assignment

CoefficientAssignment object

Coefficient assignment, returned as a CoefficientAssignment Properties object.

More About

d Coefficient When m is Nonzero

The d coefficient takes a special matrix form when m is nonzero. You must specify d as a matrix of a particular size, and not as a function handle.

d represents a damping coefficient in the case of nonzero m. To specify d, perform these two steps:

1. Call results = assembleFEMatrices(...) for the problem with your original coefficients and using d = 0. Use the default "none" method for assembleFEMatrices.

2. Take the d coefficient as a matrix of size results.M. Generally, d is either proportional to results.M, or is a linear combination of results.M and results.K.

Tips

• For eigenvalue equations, the coefficients cannot depend on the solution u or its gradient.

• You can transform a partial differential equation into the required form by using Symbolic Math Toolbox™. The pdeCoefficients (Symbolic Math Toolbox) converts a PDE into the required form and extracts the coefficients into a structure that can be used by specifyCoefficients.

  The pdeCoefficients function also can return a structure of symbolic expressions, in which case you need to use pdeCoefficientsToDouble (Symbolic Math Toolbox) to convert these expressions to double format before passing them to specifyCoefficients.

Version History

Introduced in R2016a