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specifyCoefficients

Specify coefficients in a PDE model

Description

Coefficients of a PDE

solvepde solves PDEs of the form

m2ut2+dut·(cu)+au=f

solvepdeeig solves PDE eigenvalue problems of the form

·(cu)+au=λduor·(cu)+au=λ2mu

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

example

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.

example

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

example

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

Examples

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Specify the coefficients for Poisson's equation -u=1.

solvepde addresses equations of the form

m2ut2+dut-(cu)+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)

Figure contains an axes object. The axes object contains an object of type patch.

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=y2tanh(z)/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)

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])

Figure contains an axes object. The axes object contains 4 objects of type line, text.

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)

Figure contains an axes object. The axes object contains an object of type patch.

Input Arguments

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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.

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 if the term is not part of your problem.

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

Data Types: double | function_handle
Complex Number Support: Yes

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 if the term is not part of your problem.

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

Data Types: double | function_handle
Complex Number Support: Yes

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.

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

Data Types: double | function_handle
Complex Number Support: Yes

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.

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

Data Types: double | function_handle
Complex Number Support: Yes

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.

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

Data Types: double | function_handle
Complex Number Support: Yes

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

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

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Coefficient assignment, returned as a CoefficientAssignment Properties object.

More About

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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.

See Dynamics of Damped Cantilever Beam.

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