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.

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

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.

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

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`

.

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

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

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

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.

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.

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

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

Solve the model.

Access the solution at the nodal locations.

Plot the transverse deflection of the plate.

Find the transverse deflection at the plate center.

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