3. Pressure formulation

3.1. 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, \(p\in {H}^{2}\left(\Omega \right)\). We multiply the Helmholtz equation () without a source term \(s\) by a test function \(\mathrm{\varphi }\). We integrate on \(\mathrm{\Omega }\) and we use Green’s formula. According to (), the border \(\partial \Omega\) of the domain \(\Omega\) is subdivided into two zones, a zone with imposed vibratory speed, \(\partial {\Omega }_{v}\) and a zone with imposed acoustic impedance, \(\partial {\Omega }_{Z}\). The equation obtained can be rewritten in the form:

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

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

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

3.2. Finite element discretization

In the case of classical isoparametric finite elements, the elementary integrals are \({\mathrm{K}}^{e}\), \({\mathrm{M}}^{e}\),, \({\mathrm{C}}^{e}\) and \({\mathrm{S}}^{e}\) according to the decomposition indicated by () (\({\mathrm{K}}^{e}\) is the « stiffness » matrix, \({\mathrm{M}}^{e}\) the « mass » matrix, \({C}^{e}\) the damping matrix, the damping matrix and \({\mathrm{S}}^{e}\) 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 \(m\) elementary form functions \({H}_{i}\):

(3.3)\[ {\ mathit {OM}} ^ {e} = {\ sum} = {\ sum} _ {i=1} ^ {m} {N} _ {i} {\ mathit {OM}}} _ {i}} _ {i} ^ {e}\]

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

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

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

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

Where \({p}^{e}\) is the column vector of the nodal values of the sound pressure on the element.

3.2.1. The « stiffness » matrix

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

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

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

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

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 .

3.2.2. The « mass » matrix

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

(3.8)\[ {M} _ {\ mathit {ij}}} ^ {e}} = {\ int} = {\ int} _ {\ omega} ^ {e}}\ frac {1} {{c} ^ {2}} p\ varphi\ mathit {dV}\]

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

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

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 \(c\) that occurs in () is a complex number.

3.2.3. The amortization matrix

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

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

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

\[\]

: 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}}

3.2.4. The source vector

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

\[\]

: label: eq-21

{S} _ {i} ^ {e} = {int} = {int} _ {int} _ {int} _ {int} _ {n} _ {n}varphimathit {dS} =0

It admits as a general term:

(3.11)\[ \begin{align}\begin{aligned} {S} _ {i} ^ {e} = {\ int} = {\ int} _ {\ partial {\ omega} _ {v} ^ {e}} {\ rho} _ {0} {V} _ {n} {n} {N} {N} _ {N} _ {n} {N} _ {n} {N}} {N}} {N} _ {N} {N}} {N}} {N} {N} _ {N} {N}} {N}} {N} {N}\\ {S} _ {i} ^ {e} = {\ int} = {\ int} _ {\ partial {\ omega} _ {v} ^ {e}} {\ rho} _ {0} {V} _ {n} {n} {N} {N} _ {N} _ {n} {N} _ {n} {N}} {N}} {N} _ {N} {N}} {N}} {N} {N} _ {N} {N}} {N}} {N} {N}\end{aligned}\end{align} \]