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hyperbolic

Solve hyperbolic PDE problem

Hyperbolic equation solver

Solves PDE problems of the type

d2ut2(cu)+au=f

on a 2-D or 3-D region Ω, or the system PDE problem

d2ut2(cu)+au=f

The variables c, a, f, and d can depend on position, time, and the solution u and its gradient.

hyperbolic is not recommended. Use solvepde instead.

Syntax

  • u = hyperbolic(u0,ut0,tlist,model,c,a,f,d)
    example
  • u = hyperbolic(u0,ut0,tlist,b,p,e,t,c,a,f,d)
    example
  • u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M)
    example
  • u = hyperbolic(___,rtol)
  • u = hyperbolic(___,rtol,atol)
  • u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M,___,'DampingMatrix',D)
    example
  • u = hyperbolic(___,'Stats','off')

Description

example

u = hyperbolic(u0,ut0,tlist,model,c,a,f,d) produces the solution to the FEM formulation of the scalar PDE problem

d2ut2(cu)+au=f

on a 2-D or 3-D region Ω, or the system PDE problem

d2ut2(cu)+au=f

with geometry, mesh, and boundary conditions specified in model, with initial value u0 and initial derivative with respect to time ut0. The variables c, a, f, and d in the equation correspond to the function coefficients c, a, f, and d respectively.

example

u = hyperbolic(u0,ut0,tlist,b,p,e,t,c,a,f,d) solves the problem using boundary conditions b and finite element mesh specified in [p,e,t].

example

u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M) solves the problem based on finite element matrices that encode the equation, mesh, and boundary conditions.

u = hyperbolic(___,rtol) and u = hyperbolic(___,rtol,atol) modify the solution process by passing to the ODE solver a relative tolerance rtol, and optionally an absolute tolerance atol.

example

u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M,___,'DampingMatrix',D) modifies the problem to include a damping matrix D.

u = hyperbolic(___,'Stats','off') turns off the display of internal ODE solver statistics during the solution process.

Examples

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Solve the wave equation

$$\frac{\partial^2 u}{\partial t^2} = \Delta u$$

on the square domain specified by squareg.

Create a PDE model and import the geometry.

model = createpde;
geometryFromEdges(model,@squareg);
pdegplot(model,'EdgeLabels','on')
ylim([-1.1,1.1])
axis equal

Set Dirichlet boundary conditions $u = 0$ for $x = \pm 1$, and Neumann boundary conditions

$$\nabla u\cdot\bf{n} = 0$$

for $y = \pm 1$. (The Neumann boundary condition is the default condition, so the second specification is redundant.)

applyBoundaryCondition(model,'dirichlet','Edge',[2,4],'u',0);
applyBoundaryCondition(model,'neumann','Edge',[1,3],'g',0);

Set the initial conditions

u0 = 'atan(cos(pi/2*x))';
ut0 = '3*sin(pi*x).*exp(cos(pi*y))';

Set the solution times.

tlist = linspace(0,5,31);

Give coefficients for the problem.

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

Generate a mesh and solve the PDE.

generateMesh(model,'Hmax',0.1);
u1 = hyperbolic(u0,ut0,tlist,model,c,a,f,d);
549 successful steps
69 failed attempts
1238 function evaluations
1 partial derivatives
172 LU decompositions
1237 solutions of linear systems

Plot the solution at the first and last times.

figure
pdeplot(model,'XYData',u1(:,1))
figure
pdeplot(model,'XYData',u1(:,end))

For a version of this example with animation, run pdedemo6.

Solve the wave equation

$$\frac{\partial^2 u}{\partial t^2} = \Delta u$$

on the square domain specified by squareg, using a geometry function to specify the geometry, a boundary function to specify the boundary conditions, and using initmesh to create the finite element mesh.

Specify the geometry as @squareg and plot the geometry.

g = @squareg;
pdegplot(g,'EdgeLabels','on')
ylim([-1.1,1.1])
axis equal

Set Dirichlet boundary conditions $u = 0$ for $x = \pm 1$, and Neumann boundary conditions

$$\nabla u\cdot\bf{n} = 0$$

for $y = \pm 1$. (The Neumann boundary condition is the default condition, so the second specification is redundant.)

The squareb3 function specifies these boundary conditions.

b = @squareb3;

Set the initial conditions

u0 = 'atan(cos(pi/2*x))';
ut0 = '3*sin(pi*x).*exp(cos(pi*y))';

Set the solution times.

tlist = linspace(0,5,31);

Give coefficients for the problem.

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

Create a mesh and solve the PDE.

[p,e,t] = initmesh(g);
u = hyperbolic(u0,ut0,tlist,b,p,e,t,c,a,f,d);
462 successful steps
70 failed attempts
1066 function evaluations
1 partial derivatives
156 LU decompositions
1065 solutions of linear systems

Plot the solution at the first and last times.

figure
pdeplot(p,e,t,'XYData',u(:,1))
figure
pdeplot(p,e,t,'XYData',u(:,end))

For a version of this example with animation, run pdedemo6.

Solve a hyperbolic problem using finite element matrices.

Create a model and import the BracketWithHole.stl geometry.

model = createpde();
importGeometry(model,'BracketWithHole.stl');
figure
pdegplot(model,'FaceLabels','on')
view(30,30)
title('Bracket with Face Labels')
figure
pdegplot(model,'FaceLabels','on')
view(-134,-32)
title('Bracket with Face Labels, Rear View')

Set coefficients c = 1, a = 0, f = 0.5, and d = 1.

c = 1;
a = 0;
f = 0.5;
d = 1;

Generate a mesh for the model.

generateMesh(model);

Create initial conditions and boundary conditions. The boundary condition for the rear face is Dirichlet with value 0. All other faces have the default boundary condition. The initial condition is u(0) = 0, du/dt(0) = x/2. Give the initial condition on the derivative by calculating the x-position of each node in xpts, and passing x/2.

applyBoundaryCondition(model,'Face',4,'u',0);
u0 = 0;
xpts = model.Mesh.Nodes(1,:);
ut0 = xpts(:)/2;

Create the associated finite element matrices.

[Kc,Fc,B,ud] = assempde(model,c,a,f);
[~,M,~] = assema(model,0,d,f);

Solve the PDE for times from 0 to 2.

tlist = linspace(0,5,50);
u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M);
1483 successful steps
67 failed attempts
2928 function evaluations
1 partial derivatives
277 LU decompositions
2927 solutions of linear systems

View the solution at a few times. Scale all the plots to have the same color range by using the caxis command.

umax = max(max(u));
umin = min(min(u));

subplot(2,2,1)
pdeplot3D(model,'ColorMapData',u(:,5))
caxis([umin umax])
title('Time 1/2')
subplot(2,2,2)
pdeplot3D(model,'ColorMapData',u(:,10))
caxis([umin umax])
title('Time 1')
subplot(2,2,3)
pdeplot3D(model,'ColorMapData',u(:,15))
caxis([umin umax])
title('Time 3/2')
subplot(2,2,4)
pdeplot3D(model,'ColorMapData',u(:,20))
caxis([umin umax])
title('Time 2')

The solution seems to have a frequency of one, because the plots at times 1/2 and 3/2 show maximum values, and those at times 1 and 2 show minimum values.

Solve a hyperbolic problem that includes damping. You must use the finite element matrix form to use damping.

Create a model and import the BracketWithHole.stl geometry.

model = createpde();
importGeometry(model,'BracketWithHole.stl');
figure
pdegplot(model,'FaceLabels','on')
view(30,30)
title('Bracket with Face Labels')
figure
pdegplot(model,'FaceLabels','on')
view(-134,-32)
title('Bracket with Face Labels, Rear View')

Set coefficients c = 1, a = 0, f = 0.5, and d = 1.

c = 1;
a = 0;
f = 0.5;
d = 1;

Generate a mesh for the model.

generateMesh(model);

Create initial conditions and boundary conditions. The boundary condition for the rear face is Dirichlet with value 0. All other faces have the default boundary condition. The initial condition is u(0) = 0, du/dt(0) = x/2. Give the initial condition on the derivative by calculating the x-position of each node in xpts, and passing x/2.

applyBoundaryCondition(model,'Face',4,'u',0);
u0 = 0;
xpts = model.Mesh.Nodes(1,:);
ut0 = xpts(:)/2;

Create the associated finite element matrices.

[Kc,Fc,B,ud] = assempde(model,c,a,f);
[~,M,~] = assema(model,0,d,f);

Use a damping matrix that is 10% of the mass matrix.

Damping = 0.1*M;

Solve the PDE for times from 0 to 2.

tlist = linspace(0,5,50);
u = hyperbolic(u0,ut0,tlist,Kc,Fc,B,ud,M,'DampingMatrix',Damping);
1419 successful steps
70 failed attempts
2773 function evaluations
1 partial derivatives
271 LU decompositions
2772 solutions of linear systems

Plot the maximum value at each time. The oscillations damp slightly as time increases.

plot(max(u))
xlabel('Time')
ylabel('Maximum value')
title('Maximum of Solution')

Input Arguments

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Initial condition, specified as a scalar, vector of nodal values, or text expression. The initial condition is the value of the solution u at the initial time, specified as a column vector of values at the nodes. The nodes are either p in the [p,e,t] data structure, or are model.Mesh.Nodes. For details, see Solve PDEs with Initial Conditions.

  • If the initial condition is a constant scalar v, specify u0 as v.

  • If there are Np nodes in the mesh, and N equations in the system of PDEs, specify u0 as a column vector of Np*N elements, where the first Np elements correspond to the first component of the solution u, the second Np elements correspond to the second component of the solution u, etc.

  • Give a text expression of a function, such as 'x.^2 + 5*cos(x.*y)'. If you have a system of N > 1 equations, give a text array such as

    char('x.^2 + 5*cos(x.*y)',...
        'tanh(x.*y)./(1+z.^2)')

Example: x.^2+5*cos(y.*x)

Data Types: double | char
Complex Number Support: Yes

Initial derivative, specified as a vector or text expression. The initial gradient is the value of the derivative of the solution u at the initial time, specified as a vector of values at the nodes. The nodes are either p in the [p,e,t] data structure, or are model.Mesh.Nodes. See Solve PDEs with Initial Conditions.

  • If the initial derivative is a constant value v, specify u0 as v.

  • If there are Np nodes in the mesh, and N equations in the system of PDEs, specify ut0 as a vector of Np*N elements, where the first Np elements correspond to the first component of the solution u, the second Np elements correspond to the second component of the solution u, etc.

  • Give a text expression of a function, such as 'x.^2 + 5*cos(x.*y)'. If you have a system of N > 1 equations, use a text array such as

    char('x.^2 + 5*cos(x.*y)',...
        'tanh(x.*y)./(1+z.^2)')

For details, see Solve PDEs with Initial Conditions.

Example: p(1,:).^2+5*cos(p(2,:).*p(1,:))

Data Types: double | char
Complex Number Support: Yes

Solution times, specified as a real vector. The solver returns the solution to the PDE at the solution times.

Example: 0:0.2:4

Data Types: double

PDE model, specified as a PDEModel object.

Example: model = createpde(1)

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

d2ut2(cu)+au=f

or in the system of PDEs

d2ut2(cu)+au=f

You can specifyc in various ways, detailed in c Coefficient for Systems. See also Specify Scalar PDE Coefficients in Character Form, Specify 2-D Scalar Coefficients in Function Form, and Specify 3-D PDE Coefficients in Function Form.

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

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

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

d2ut2(cu)+au=f

or in the system of PDEs

d2ut2(cu)+au=f

There are a wide variety of ways of specifying a, detailed in a or d Coefficient for Systems. See also Specify Scalar PDE Coefficients in Character Form, Specify 2-D Scalar Coefficients in Function Form, and Specify 3-D PDE Coefficients in Function Form.

Example: 2*eye(3)

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

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

d2ut2(cu)+au=f

or in the system of PDEs

d2ut2(cu)+au=f

You can specifyf in various ways, detailed in f Coefficient for Systems. See also Specify Scalar PDE Coefficients in Character Form, Specify 2-D Scalar Coefficients in Function Form, and Specify 3-D PDE Coefficients in Function Form.

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

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

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

d2ut2(cu)+au=f

or in the system of PDEs

d2ut2(cu)+au=f

You can specifyd in various ways, detailed in a or d Coefficient for Systems. See also Specify Scalar PDE Coefficients in Character Form, Specify 2-D Scalar Coefficients in Function Form, and Specify 3-D PDE Coefficients in Function Form.

Example: 2*eye(3)

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

Boundary conditions, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a file name.

Example: b = 'circleb1' or equivalently b = @circleb1

Data Types: double | char | function_handle

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.

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

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

Data Types: double

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.

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

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

Data Types: double

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.

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

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

Data Types: double

Stiffness matrix, specified as a sparse matrix or as a full matrix. See Elliptic Equations. Typically, Kc is the output of assempde.

Load vector, specified as a vector. See Elliptic Equations. Typically, Fc is the output of assempde.

Dirichlet nullspace, returned as a sparse matrix. See Algorithms. Typically, B is the output of assempde.

Dirichlet vector, returned as a vector. See Algorithms. Typically, ud is the output of assempde.

Mass matrix. specified as a sparse matrix or a full matrix. See Elliptic Equations.

To obtain the input matrices for pdeeig, hyperbolic or parabolic, run both assema and assempde:

[Kc,Fc,B,ud] = assempde(model,c,a,f);
[~,M,~] = assema(model,0,d,f);

    Note:   Create the M matrix using assema with d, not a, as the argument before f.

Data Types: double
Complex Number Support: Yes

Relative tolerance for ODE solver, specified as a positive real.

Example: 2e-4

Data Types: double

Absolute tolerance for ODE solver, specified as a positive real.

Example: 2e-7

Data Types: double

Damping matrix, specified as a matrix. D has the same size as the stiffness matrix Kc or the mass matrix M. When you include D, hyperbolic solves the following ODE for the variable v:

BTMBd2vdt2+BTDBdvdt+Kv=F

with initial condition u0 and initial derivative ut0. Then hyperbolic returns the solution u = B*v + ud.

For an example using D, see Dynamics of a Damped Cantilever Beam.

Example: alpha*M + beta*K

Data Types: double
Complex Number Support: Yes

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example:

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Display ODE solver statistics, specified as 'on' or 'off'. Suppress the display by setting Stats to 'off'.

Example: x = hyperbolic(u0,ut0,tlist,model,c,a,f,d,'Stats','off')

Data Types: char

Output Arguments

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PDE solution, returned as a matrix. The matrix is Np*N-by-T, where Np is the number of nodes in the mesh, N is the number of equations in the PDE (N = 1 for a scalar PDE), and T is the number of solution times, meaning the length of tlist. The solution matrix has the following structure.

  • The first Np elements of each column in u represent the solution of equation 1, then next Np elements represent the solution of equation 2, etc. The solution u is the value at the corresponding node in the mesh.

  • Column i of u represents the solution at time tlist(i).

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

To plot the solution, use pdeplot for 2-D geometry, or see Plot 3-D Solutions and Their Gradients.

More About

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Algorithms

hyperbolic internally calls assema, assemb, and assempde to create finite element matrices corresponding to the problem. It calls ode15s to solve the resulting system of ordinary differential equations. For details, see Hyperbolic Equations.

See Also

Introduced before R2006a

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