Benchmark solution ===================== Calculation method ----------------- The problem is spherically symmetric. We therefore place ourselves in the :math:`(r,\mathrm{\theta },\mathrm{\varphi })` frame of reference. We consider the case where :math:`{p}_{\mathit{imp}}` does not depend on :math:`\mathrm{\theta }`, nor on :math:`\mathrm{\varphi }`. This hypothesis allows us to express the analytical solution for the problem in a harmonic regime and semi-analytic for the transient problem by exploiting the symmetry of the problem. In fact, by the symmetry in the problem, the structural displacement is written as: .. math:: :label: eq-1 {u} _ {s} (r,\ mathrm {\ theta},\ mathrm {\ varphi}, t) =u (r, t) {e} _ {r} And the sound pressure in the fluid: .. math:: :label: eq-2 {p} _ {f} (r,\ mathrm {\ theta},\ mathrm {\ varphi}, t) =p (r, t) =p (r, t) The linearized Green-Lagrange strain tensor is therefore given by: .. math:: :label: eq-3 \ mathrm {\ epsilon} (r, t) =\ frac {\ partial u (r, t)} {\ partial r} (r, t) {e} _ {r}\ otimes {e} _ {r} _ {r} +\ frac {u (r, t)} +\ frac {u (r, t)} {\ frac {u, t)} {r, t)} {\ frac {u (r, t)} {r} _ {\ mathrm {\ theta}} + {e}} _ {\ mathrm {\ varphi}}\ otimes {e} _ {\ mathrm {\ varphi}}}\ right) With an isotropic elastic material, the Cauchy stress is then equal to: .. math:: : label: eq-4 \ mathrm {\ sigma} (r, t) =\ left ((\ mathrm {\ lambda} +2\ mathrm {\ mu})\ frac {\ partial u (r, t)} {\ partial r}} {\ partial r} +2\ mathrm {\ lambda} +2\ mathrm {\ lambda}\ lambda}\ frac {\ lambda}\ frac {u (r, t)} {r}\ otimes {e} _ {r} +\ left (\ mathrm {\ lambda}\ frac {\ partial u (r, t)} {\ partial r} + (\ mathrm {\ lambda} +2\ mathrm {\ mu})\ frac {\ mu})\ frac {u (r, t)} {r, t)} {r, t)} {r}\ right)\ left ({e} _ {\ mathrm {\ theta}}}\ otimes {e} _ {\ mathrm {\ theta}} + {e}} _ {\ mathrm {\ varphi}}\ otimes {e} _ {\ mathrm {\ varphi}}\ right) :math:`\mathrm{\lambda }` and :math:`\mathrm{\mu }` being the Lamé coefficients. The expression for the conservation of momentum allows us to write: .. math:: : label: eq-5 \ frac {{r} ^ {2}} {{c}} {{c} _ {s} ^ {2}}\ frac {{\ partial} ^ {2}} u (r, t)} {\ partial {t} ^ {2}}} - {r} {2}} - {r}}} - {r} {2}}} -r\ frac {\ partial u (r, t)}} {\ partial r} +2u (r, t) =0 With the speed of sound propagation in structure :math:`{c}_{s}` which is equal to: .. math:: :label: eq-6 {c} _ {s} ^ {2} =\ frac {\ mathrm {\ lambda} +2\ mathrm {\ mu}} {{\ mathrm {\ rho}}} _ {s}} :math:`\mathrm{\lambda }` and :math:`\mathrm{\mu }` being the Lamé coefficients. .. math:: :label: eq-7 \ frac {1} {{c} _ {f} ^ {2}}}\ frac {{\ partial} ^ {2} p} {\ partial {t} ^ {2}} (t, x) -\ mathrm {\ Delta} p (t, x) =0 In spherical coordinates, the sound pressure propagation equation is written as: .. math:: :label: eq-8 \ frac {1} {{c} _ {f} ^ {2}}}\ frac {{\ partial} ^ {2} p} {\ partial {t} ^ {2}} (r, t) -\ frac {{\ partial}} ^ {2}}}\ frac {2}} {\ partial} ^ {2}}} (r, t) -\ frac {2} {r} {r} {r}\ frac {\ partial} ^ {2}}} (r, t) -\ frac {2} {r} {r}}\ frac {\ partial} {\ partial}} partial p} {\ partial r} (r, t) =0 The condition of continuity of the normal stress on the surface :math:`{\mathrm{\Gamma }}_{\mathit{ext}}` is written in the form of a scalar equation: .. csv-table:: ":math:`(\mathrm{\lambda }+2\mathrm{\mu }){\frac{\partial u(t)}{\partial r}", "}_{r={R}_{\mathit{ext}}}+2\mathrm{\lambda }{\frac{u(t)}{{R}_{\mathit{ext}}}", "}_{r={R}_{\mathit{ext}}}=-{p(t)", "}_{r={R}_{\mathit{ext}}}` ", "(9)" The condition for the continuity of normal acceleration on surface :math:`{\mathrm{\Gamma }}_{\mathit{ext}}` is written as: .. csv-table:: ":math:`{\frac{{\partial }^{2}u(t)}{\partial {t}^{2}}", "}_{{R}_{\mathit{ext}}}=-{\frac{1}{{\mathrm{\rho }}_{f}}\frac{\partial p(t)}{\partial r}", "}_{{R}_{\mathit{ext}}}` ", "(10)" The pressure is imposed on surface :math:`{\mathrm{\Gamma }}_{\text{int}}` and is written as: .. csv-table:: ":math:`(\mathrm{\lambda }+2\mathrm{\mu }){\frac{\partial u(t)}{\partial r}", "}_{r={R}_{\text{int}}}+2\mathrm{\lambda }{\frac{u(t)}{{R}_{\mathit{ext}}}", "}_{r={R}_{\text{int}}}=-{p}_{\mathit{imp}}(t)` ", "(11)" Finally, we apply the Sommerfeld condition (non-reflection of waves) on the outer surface :math:`{\mathrm{\Gamma }}_{\mathit{bgt}}`: .. math:: :label: eq-12 \ underset {r\ to\ mathrm {\ infty}} {\ mathrm {lim}}}\ left (\ frac {1} {{c} _ {f}}\ frac {\ partial p} {\ partial p} {\ partial p} {\ partial p} {\ partial r}}\ frac {\ partial p} {\ partial r}} (r, t)\ right) =0 Harmonic resolution ------------------------ We are interested here in soliciting the form :math:`{p}_{\mathit{imp}}(t)={P}_{\mathit{imp}}{e}^{i\mathrm{\omega }t}` and we are looking for the solution of the form :math:`u(r,t)=U(r){e}^{i\mathrm{\omega }t}` and :math:`p(r,t)=P(r){e}^{i\mathrm{\omega }t}`. In this form, partial time derivatives are trivial. For example: .. math:: :label: eq-13 u (r, t) =U (r) {e} ^ {i\ mathrm {\ omega} t}\ to\ frac {\ partial u} {\ partial t} =i\ mathrm {\ omega} U (r) {\ omega} U (r) {e} ^ {e} ^ {e} ^ {e} ^ {e} ^ {e} ^ {i\ mathrm {\ omega} t}\ to\ frac {{\ partial} ^ {2} u} {\ partial t} u} {\ partial t} ^ {2}} =-i {\ mathrm {\ omega}}} ^ {2} U (r) {e} ^ {i\ mathrm {\ omega} t} The problem comes down to finding :math:`U` and :math:`P` which are solutions to the following problem: .. math:: :label: eq-14 {r} ^ {2}\ frac {{d}} ^ {2} U} {{\ mathit {dr}} ^ {2}} +2r\ frac {dU} {\ mathit {dr}}} +\ left (\ left (\ frac {{\ mathrm {\ omega}}} {{\ mathrm {\ omega}}} {\ mathrm {\ omega}}} ^ {2}}} {r} ^ {2}}} +\ left (\ frac {{\ mathrm {\ omega}}} ^ {2}}} {r} ^ {2}}} {r} ^ {2}} {r} ^ {2}}} +\ left (\ frac {{d}}}} +\ left (\ frac {{d}}\ right) U=0 And: .. math:: :label: eq-15 \ frac {{d} ^ {2} P} {{\ mathit {dr}} {{\ mathit {dr}}}} ^ {2}} +\ frac {\ mathit {dP}} {\ mathit {dr}}} {\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {\ mathit {dr}}} +\ frac {{\ mathit {dr} The solution to the problem () is written as: .. math:: :label: eq-16 U (r) =A (\ mathrm {\ omega}) {j} _ {1}\ left (\ frac {\ mathrm {\ omega}} {{c} _ {s}} r\ right) +B (\ mathrm {\ omega}) {\ mathrm {\ omega}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}) +B (\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\} r\ right) Where :math:`{j}_{1}` and :math:`{y}_{1}` are, respectively, the first-order spherical Bessel function of the first type and the first-order spherical Bessel function of the second type. The solution to the problem () is written as: .. math:: :label: eq-17 P (r) =C (\ mathrm {\ omega})\ frac {{e} ^ {-i\ mathrm {\ omega} r/ {c} _ {f}}} {r}} {r} +D (\ mathrm {\ omega})\ frac {\ omega})\ frac {{e} ^ {e} ^ {i\ mathrm {\ omega} r/ {c}} _ {f}}} {f}} {r}} {r} To find the values :math:`A(\mathrm{\omega })`, :math:`B(\mathrm{\omega })`, :math:`C(\mathrm{\omega })`, and :math:`D(\mathrm{\omega })`. Boundary conditions are used. In equation (), the term in :math:`\frac{{e}^{-i\mathrm{\omega }r/{c}_{f}}}{r}` corresponds to the wave propagating radially to infinity while the other term :math:`\frac{{e}^{i\mathrm{\omega }r/{c}_{f}}}{r}` is the reflection of the wave (towards the source). As we impose a non-reflection condition, on the outer edge, this second contribution is zero and we therefore have :math:`D(\mathrm{\omega })=0`. Reference quantities and results ----------------------------------- To calculate the solution, you need to have an estimate of the Bessel functions. For that we use scipy, (special functions). The displacement is recovered at two points: Uint on the surface of the structure :math:`{\mathrm{\Gamma }}_{\text{int}}`, Uext on the surface IFS :math:`{\mathrm{\Gamma }}_{\text{ext}}`. The pressure is recorded Pext on the surface IFS :math:`{\mathrm{\Gamma }}_{\text{ext}}` and Pbgt on the outer surface :math:`{\mathrm{\Gamma }}_{\text{bgt}}` *.* +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |Real part |Imaginary part | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020100000280000001E08F60D4CE2EB6CCF0.png + .. image:: images/1000020100000280000001E0D87C3C0FB7AA765E.png + | :width: 3.3681in | :width: 3.3681in | + :height: 2.5256in + :height: 2.5256in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020100000280000001E0ADC02A993FFE55DB.png + .. image:: images/1000020100000280000001E07EEAA6F6D400B7E9.png + | :width: 3.3681in | :width: 3.3681in | + :height: 2.5256in + :height: 2.5256in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020100000280000001E07C7FD3DC04E4B3B2.png + .. image:: images/1000020100000280000001E00605C02F6BB9ADF2.png + | :width: 3.3681in | :width: 3.3681in | + :height: 2.5256in + :height: 2.5256in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020100000280000001E0E999A63DBADD9688.png + .. image:: images/1000020100000280000001E0A87DDBC86B4AE8B5.png + | :width: 3.3681in | :width: 3.3681in | + :height: 2.5256in + :height: 2.5256in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ Uncertainties about the solution ---------------------------- None, the solution is analytical.