Pressure formulation
=======================




Mathematical expression of the problem
-----------------------------------

The standard procedure for solving the problem with classical isoparametric finite elements is used. We assume the solution of the problem is fairly regular, :math:`p\in {H}^{2}\left(\Omega \right)`. We multiply the Helmholtz equation () without a source term :math:`s` by a test function :math:`\mathrm{\varphi }`. We integrate on :math:`\mathrm{\Omega }` and we use Green's formula. According to (), the border :math:`\partial \Omega` of the domain :math:`\Omega` is subdivided into two zones, a zone with imposed vibratory speed, :math:`\partial {\Omega }_{v}` and a zone with imposed acoustic impedance, :math:`\partial {\Omega }_{Z}`. The equation obtained can be rewritten in the form:



.. math::
 :label: eq-10

    {\ int} _ {\ mathrm {\ Omega}}\ mathrm {grad} (p)\ cdot\ mathrm {grad} (\ mathrm {\ varphi})\ mathit {dV} - {\ int} - {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ mathrm {\ omega}}\ frac {\ mathrm {\ omega}}} ^ {2}} {{c}} ^ {2}} {{c}} ^ {2}} {{c}} ^ {2}} {{c}} ^ {2}} {{c}} ^ {2}} {{c}} ^ {2}}}} p\ mathrm {\ varphi}\ mathit {dV} +j {\ int} +j {\ int} _ {\ mathrm {\ omega}} _ {Z}}\ frac {{\ mathrm {\ rho}}} _ {0}\ mathrm {\ rho}}} _ {0}\ mathrm {\ omega}} {Z} p\ mathrm {\ varphi}}\ frac {{\ mathrm {\ s} +j {\ rho}} _ {0}\ mathrm {\ omega}} {Z}}\ mathrm {\ varphi}}\ frac {{\ mathrm {\ rho}}}\ frac {{\ mathrm {\ rho}}} {\ rho}} int} _ {\ partial {\ mathrm {\ omega}}} _ {\ omega}} _ {\ mathrm {\ rho}} _ {0}\ mathrm {\ omega} {V} {V} _ {n} _ {n}\ mathrm {\ omega}} _ {n}\ mathrm {\ varphi}}\ mathit {dS} =0


where :math:`\mathit{dV}` represents a differential volume element in :math:`\Omega` and :math:`\mathit{dS}` represents a surface element out of :math:`\partial \Omega`. The particle vibratory speed is then determined by:



.. math::
 :label: eq-11

    \ mathrm {v} =\ frac {j} {{\ mathrm {\ rho}} _ {0}\ mathrm {\ omega}}}\ mathrm {grad} (p)





Finite element discretization
---------------------------------

In the case of classical isoparametric finite elements, the elementary integrals are :math:`{\mathrm{K}}^{e}`, :math:`{\mathrm{M}}^{e}`,, :math:`{\mathrm{C}}^{e}` and :math:`{\mathrm{S}}^{e}` according to the decomposition indicated by () (:math:`{\mathrm{K}}^{e}` is the "stiffness" matrix, :math:`{\mathrm{M}}^{e}` the "mass" matrix, :math:`{C}^{e}` the damping matrix, the damping matrix and :math:`{\mathrm{S}}^{e}` the source vector). Two of them come from volume integrals, the other two are the result of integrals respectively on a vibrating surface and on a surface with imposed impedance. We will assume that the global coordinates of an element can be written using the data of :math:`m` elementary form functions :math:`{H}_{i}`:


.. math::
 :label: eq-12

    {\ mathit {OM}} ^ {e} = {\ sum} = {\ sum} _ {i=1} ^ {m} {N} _ {i} {\ mathit {OM}}} _ {i}} _ {i} ^ {e}


In addition, basic functions :math:`{N}_{i}` are given to describe elementary sound pressure. The pressure inside an element can be written as:


.. math::
 :label: eq-13

    {p} ^ {e} (x, y, z) = {\ sum} _ {i=1} ^ {m} {N} _ {i} {p} _ {i} {p} _ {i} ^ {e}


Where :math:`{p}_{i}^{e}` is the sound pressure at node :math:`i` of element :math:`e`. In the case of isoparametric finite elements, the base functions :math:`{N}_{i}` are equal to the functions of the form :math:`{H}_{i}`. For each element in the field, the finite element problem in sound pressure is written:


.. math::
 :label: eq-14

    \ left ({\ mathrm {K}}} ^ {e}} - {\ mathrm {\ omega}}} ^ {2} {\ mathrm {M}} ^ {e} +j\ mathrm {\ omega}} {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ mathrm {C}}} {\ mathrm {C}}}} ^ {e}\ right) {p} ^ {e} =-j\ mathrm {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ mathrm {C}}} {\ mathrm {C}}}} ^ {e}\ right) {p} ^ {e} =-j\ mathrm {}} ^ {e}


Where :math:`{p}^{e}` is the column vector of the nodal values of the sound pressure on the element.




The "stiffness" matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The "stiffness" matrix :math:`{\mathrm{K}}^{e}` corresponds to the calculation of the first term of (), i.e.:



.. math::
 :label: eq-15

    {\ int} _ {{\ mathrm {\ omega}}} ^ {e}}}\ left (\ mathrm {grad} (p)\ cdot\ mathrm {grad} (\ mathrm {\ varphi})}} ^ {e}}}\ left (\ mathrm {\ varphi})\ right)\ mathit {dV}


It corresponds to the kinetic energy of the acoustic fluid. It accepts as a general term:


.. math::
 :label: eq-16

    {K} _ {\ mathit {ij}}} ^ {e}} = {\ int} = {\ int} _ {\ omega} ^ {e}}\ left (\nabla {N} _ {i}\nabla {N}}\nabla {N} _ {N} _ {j}\ right)\ mathrm {.} \ mathit {dV}


**Note:**

*because of the choice of pressure as the main variable and the second derivation in time,* *the* *first term of () corresponds to* kinetic energy\ *.* *For a practical reason it is nevertheless called a "stiffness" matrix .*




The "mass" matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~

The "mass" matrix :math:`{\mathrm{M}}^{e}` corresponds to the calculation of the second term of (), i.e.:



.. math::
 :label: eq-17

    {M} _ {\ mathit {ij}}} ^ {e}} = {\ int} = {\ int} _ {\ omega} ^ {e}}\ frac {1} {{c} ^ {2}} p\ varphi\ mathit {dV}


It corresponds to the compressibility energy of the acoustic fluid. It accepts as a general term:


.. math::
 :label: eq-18

    {M} _ {\ mathit {ij}}} ^ {e}} = {\ int} = {\ int} _ {\ omega} ^ {e}}\ frac {1} {{c} ^ {2}} {N}} {N}} {N} _ {j}\ mathrm {.}} {N} \ mathit {dV}


**Note 1:**

*Because of the choice of pressure as the main variable and the second derivation in time,* * *the second term of () corresponds to* *elastic energy.* *For practical reasons, it is nevertheless called a "mass" matrix. .*

**Note2:**

*In the case of propagation* *in a hysteretic viscoacoustic environment, the speed of the sound* :math:`c` *that occurs in () is a complex number.*




The amortization matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The damping matrix :math:`{\mathrm{C}}^{e}` corresponds to the calculation of the third term of (), i.e.:



.. math::
 :label: eq-19

    {C} _ {\ mathit {ij}} ^ {e}} = {\ int} = {\ int} _ {\ partial {\ omega} _ {Z} ^ {e}}\ frac {{\ rho}} _ {0}} _ {0}}} {0}}} {Z} {Z} p\ varphi\ mathit {dS}


It accepts as a general term, on the borders concerned:


.. math::
: label: eq-20

    {C} _ {\ mathit {ij}}} ^ {e} = {\ int} = {\ int} _ {\ partial {\ omega} _ {Z} ^ {e}}\ frac {{\ rho} _ {0}} _ {0}}} {Z}} {Z}} {Z} {Z} {0}}} {Z}} {Z} {0}}} {Z} {0}}} {Z} {Z}} {0}} {Z}} {Z} {0}}} {Z} {N}}





The source vector
~~~~~~~~~~~~~~~~~~

The source vector :math:`{\mathrm{S}}^{e}` corresponds to the calculation of the last term of (), i.e.:



.. math::
: label: eq-21

    {S} _ {i} ^ {e} = {\ int} = {\ int} _ {\ int} _ {\ int} _ {\ int} _ {n} _ {n}\ varphi\ mathit {dS} =0


It admits as a general term:


.. math::
 :label: eq-22

    {S} _ {i} ^ {e} = {\ int} = {\ int} _ {\ partial {\ omega} _ {v} ^ {e}} {\ rho} _ {0} {V} _ {n} {n} {N} {N} _ {N} _ {n} {N} _ {n} {N}} {N}} {N} _ {N} {N}} {N}} {N} {N} _ {N} {N}} {N}} {N} {N}

    {S} _ {i} ^ {e} = {\ int} = {\ int} _ {\ partial {\ omega} _ {v} ^ {e}} {\ rho} _ {0} {V} _ {n} {n} {N} {N} _ {N} _ {n} {N} _ {n} {N}} {N}} {N} _ {N} {N}} {N}} {N} {N} _ {N} {N}} {N}} {N} {N}