pdeCoefficientsToDouble

Convert symbolic PDE coefficients to double format

Since R2021a

Syntax

coeffs = pdeCoefficientsToDouble(symCoeffs)

coeffs = pdeCoefficientsToDouble(symCoeffs,u)

Description

coeffs = pdeCoefficientsToDouble(symCoeffs) converts the symbolic objects of the structure symCoeffs to double-precision numbers or function handles. The output is a structure coeffs that can then be used to define the coefficients of a PDE model by calling specifyCoefficients in Partial Differential Equation Toolbox™.

The structure coeffs contains the coefficients m, d, c, a, and f for the PDE system with the form

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

that can be solved in Partial Differential Equation Toolbox. For more information, see Equations You Can Solve Using Partial Differential Equation Toolbox (Partial Differential Equation Toolbox).

coeffs = pdeCoefficientsToDouble(symCoeffs,u) converts the symbolic objects of the structure symCoeffs to double-precision numbers or function handles for the coefficients of a PDE system with the dependent variables u.

Examples

Extract Coefficients From a Symbolic PDE

Create a symbolic PDE that represents the Laplacian of the variable u(x,y).

syms u(x,y) f
pdeeq = laplacian(u,[x y]) == f

pdeeq(x, y) = 
$$\frac{{\partial }^2}{\partial x^2}\mathrm{ }u\left(x,y\right)+\frac{{\partial }^2}{\partial y^2}\mathrm{ }u\left(x,y\right)=f$$

Extract the coefficients of the PDE as symbolic expressions and display their values.

symCoeffs = pdeCoefficients(pdeeq,u,'Symbolic',true);
structfun(@disp,symCoeffs)

$$0$$

$$0$$

$$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$$

$$f$$

$$0$$

pdeCoefficients converts the symbolic PDE into a scalar PDE equation of the form

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

and extract the coefficients a, c, m, d, and f into the structure symCoeffs.

Choose a value for f. Since symCoeffs are symbolic objects, use pdeCoefficientsToDouble to convert the coefficients to double data type. The coefficients with double data type are valid inputs for the specifyCoefficients function in the PDE Toolbox.

symCoeffs = subs(symCoeffs,f,-3);
coeffs = pdeCoefficientsToDouble(symCoeffs)

coeffs = struct with fields:
    a: 0
    c: [4x1 double]
    m: 0
    d: 0
    f: -3

System of Several PDEs

Solve a system of two second-order PDEs. You can solve the PDE system by extracting the PDE coefficients symbolically using pdeCoefficients, converting the coefficients to double-precision numbers using pdeCoefficientsToDouble, and specifying the coefficients in the PDE model using specifyCoefficients.

The system of PDEs represents the deflection of a clamped structural plate under a uniform pressure load. The system of PDEs with the dependent variables $$u_1$$ and $$u_2$$ is given by

$$-{\nabla }^2{u}_1+u_2=0$$,

$$-D{\nabla }^2{u}_2=p$$,

where $$D$$ is the bending stiffness of the plate given by

$$D=\frac{E{h}^3}{12(1-{\nu }^2)}$$,

and $$E$$ is the modulus of elasticity, $$\nu$$ is Poisson's ratio, $$h$$ is the plate thickness, $$u_1$$ is the transverse deflection of the plate, and $$p$$ is the pressure load.

Create a PDE model for the system of two equations.

model = createpde(2);

Create a square geometry. Specify the side length of the square. Then include the geometry in the PDE model.

len = 10.0;
gdm = [3 4 0 len len 0 0 0 len len]';
g = decsg(gdm,'S1',('S1')');
geometryFromEdges(model,g);

Specify the values of the physical parameters of the system. Let the external pressure $$p$$ be a symbolic variable pres that can take any value.

E = 1.0e6;
h_thick = 0.1;
nu = 0.3;
D = E*h_thick^3/(12*(1 - nu^2));
syms pres

Declare the PDE system as a system symbolic equations. Extract the coefficients of the PDE and return them in symbolic form.

syms u1(x,y) u2(x,y)
pdeeq = [-laplacian(u1) + u2; -D*laplacian(u2) - pres];
symCoeffs = pdeCoefficients(pdeeq,[u1 u2],'Symbolic',true)

symCoeffs = struct with fields:
    m: 0
    a: [2x2 sym]
    c: [4x4 sym]
    f: [2x1 sym]
    d: 0

Display the coefficients m, a, c, f, and d.

structfun(@disp,symCoeffs)

$$0$$

$$\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$$

$$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \frac{25000}{273}& 0\\ 0& 0& 0& \frac{25000}{273}\end{array}\right)$$

$$\left(\begin{array}{c}0\\ \mathrm{pres}\end{array}\right)$$

$$0$$

Substitute a value for pres using the subs function. Since the outputs of subs are symbolic objects, use the pdeCoefficientsToDouble function to convert the coefficients to the double data type, which makes them valid inputs for the PDE Toolbox.

symCoeffs = subs(symCoeffs,pres,2);
coeffs = pdeCoefficientsToDouble(symCoeffs)

coeffs = struct with fields:
    a: [4x1 double]
    c: [16x1 double]
    m: 0
    d: 0
    f: [2x1 double]

Specify the PDE coefficients for the PDE model.

specifyCoefficients(model,'m',coeffs.m,'d',coeffs.d, ...
    'c',coeffs.c,'a',coeffs.a,'f',coeffs.f);

Specify spring stiffness. Specify boundary conditions by defining distributed springs on all four edges.

k = 1e7;
bOuter = applyBoundaryCondition(model,'neumann','Edge',(1:4), ...
    'g',[0 0],'q',[0 0; k 0]);

Specify the mesh size of the geometry and generate a mesh for the PDE model.

hmax = len/20;
generateMesh(model,'Hmax',hmax);

Solve the model.

res = solvepde(model);

Access the solution at the nodal locations.

u = res.NodalSolution;

Plot the transverse deflection of the plate.

numNodes = size(model.Mesh.Nodes,2);
figure;
pdeplot(model,'XYData',u(1:numNodes),'contour','on')
title 'Transverse Deflection'

Find the transverse deflection at the plate center.

wMax = min(u(1:numNodes,1))

wMax = 
-0.2763

Compare the result with the deflection at the plate center computed analytically.

pres = 2;
wMax = -.0138*pres*len^4/(E*h_thick^3)

wMax = 
-0.2760

Specify Dependent Variables of PDE System

Since R2023a

Create a PDE system of four equations with four dependent variables, $\mathit{A}$, $\mathit{E}$, $\mathit{P}$, and $\mathit{T}$.

syms A(x,y,z,t) E(x,y,z,t) P(x,y,z,t) T(x,y,z,t)
eqn1 = diff(A,t) == -A*E*(-exp(T));
eqn2 = diff(P,t) == -P*E*(-exp(T));
eqn3 = diff(E,t) == -A*E*(-exp(T)) - P*E*(-exp(T));
eqn4 = diff(T,t) == laplacian(T,[x y z]) + A*E*(-exp(T)) + P*E*(-exp(T));
pdeeq = [eqn1; eqn2; eqn3; eqn4]

pdeeq(x, y, z, t) = 
$$\begin{array}{l}\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }A\left(x,y,z,t\right)={\sigma }_2\\ \frac{\partial }{\partial t}\mathrm{ }P\left(x,y,z,t\right)={\sigma }_1\\ \frac{\partial }{\partial t}\mathrm{ }\text{E}\left(x,y,z,t\right)={\sigma }_2+{\sigma }_1\\ \frac{\partial }{\partial t}\mathrm{ }T\left(x,y,z,t\right)=\frac{{\partial }^2}{\partial x^2}\mathrm{ }T\left(x,y,z,t\right)+\frac{{\partial }^2}{\partial y^2}\mathrm{ }T\left(x,y,z,t\right)+\frac{{\partial }^2}{\partial z^2}\mathrm{ }T\left(x,y,z,t\right)-{\sigma }_2-{\sigma }_1\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1={\mathrm{e}}^{T\left(x,y,z,t\right)}\hspace{0.17em}\text{E}\left(x,y,z,t\right)\hspace{0.17em}P\left(x,y,z,t\right)\\ \\ \mathrm{ }{\sigma }_2={\mathrm{e}}^{T\left(x,y,z,t\right)}\hspace{0.17em}A\left(x,y,z,t\right)\hspace{0.17em}\text{E}\left(x,y,z,t\right)\end{array}$$

Extract the coefficients of the PDE system as symbolic expressions. Specify the variable u to represent the dependent variables.

u = [A E P T];
symCoeffs = pdeCoefficients(pdeeq,u,'Symbolic',true)

symCoeffs = struct with fields:
    m: [4x4 sym]
    a: [4x4 sym]
    c: [12x12 sym]
    f: [4x1 sym]
    d: [4x4 sym]

Display the coefficients m, a, c, f, and d.

structfun(@disp,symCoeffs)

$$\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$

$$\begin{array}{l}\left(\begin{array}{cccc}-{\sigma }_2& 0& 0& 0\\ 0& {\sigma }_1& 0& 0\\ -{\sigma }_2& {\sigma }_1& 0& 0\\ {\sigma }_2& {\mathrm{e}}^{T\left(x,y,z,t\right)}\hspace{0.17em}P\left(x,y,z,t\right)& 0& 0\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=-{\mathrm{e}}^{T\left(x,y,z,t\right)}\hspace{0.17em}P\left(x,y,z,t\right)\\ \\ \mathrm{ }{\sigma }_2={\mathrm{e}}^{T\left(x,y,z,t\right)}\hspace{0.17em}\text{E}\left(x,y,z,t\right)\end{array}$$

$$\left(\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right)$$

$$\left(\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right)$$

$$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$

Convert the coefficients to the double data type so that they are valid inputs for the specifyCoefficients function in Partial Differential Equation Toolbox. Because the a coefficient in symCoeffs contains the dependent variables and is not constant, calling pdeCoefficientsToDouble(symCoeffs) without the second input argument can return an error. Instead, specify the second argument as the dependent variables u when using pdeCoefficientsToDouble.

coeffs = pdeCoefficientsToDouble(symCoeffs,u)

coeffs = struct with fields:
    a: @makeCoefficient/coefficientFunction
    c: [144x1 double]
    m: 0
    d: [16x1 double]
    f: 0

Input Arguments

symCoeffs — Coefficients of PDE in symbolic form

structure of symbolic expressions

Coefficients of PDE in symbolic form, specified as a structure of symbolic expressions.

u — Dependent variables of PDE

symbolic function

Since R2023a

Dependent variables of PDE, specified as a symbolic function. The argument u must contain stationary or time-dependent variables in two or three dimensions.

Example: syms E(x,y,z,t) B(x,y,z,t); u = [E B];

Output Arguments

coeffs — Coefficients of PDE operating on doubles

structure of doubles and function handles

Coefficients of PDE operating on doubles, returned as a structure of double-precision numbers and function handles as required by the specifyCoefficients function. The fields of the structure are a, c, m, d, and f. For details on interpreting the coefficients in the format required by Partial Differential Equation Toolbox, see:

• c Coefficient for specifyCoefficients (Partial Differential Equation Toolbox)

• m, d, or a Coefficient for specifyCoefficients (Partial Differential Equation Toolbox)

• f Coefficient for specifyCoefficients (Partial Differential Equation Toolbox)

When pdeCoefficientsToDouble returns a coefficient as a function handle, the function handle takes two structures as input arguments, location and state, and returns double-precision output. The function handle is displayed as @makeCoefficient/coefficientFunction in the Command Window. To display the formula of the function handle in coeffs.f in symbolic form, use coeffs.f('show').

In some cases, not all generated coefficients can be used by specifyCoefficients. For example, the d coefficient must take a special matrix form when m is nonzero. For more details, see d Coefficient When m Is Nonzero (Partial Differential Equation Toolbox).

Version History

Introduced in R2021a

R2023a: Specify dependent variables when converting PDE coefficients

You can specify the dependent variables u of a PDE system as the second input argument of pdeCoefficientsToDouble when converting symbolic coefficients of the PDE system to double-precision numbers or function handles. For example, see Specify Dependent Variables of PDE System.