# Numerical Analyses of Ultrasonic Atomization Utilizing Acoustic Effects of a Beam Diaphragm
## 3 Numerical Analyses  
### Finite Element Formulation of Harmonic Response of Linear Elastic Materials  
As preparation of finite element formulation for interaction between the diaphragm and water film in the next section, a summary of the general finite element formulation for dynamics of solids, which is applicable to the diaphragm, is presented in this section. For static problems of linear elastic materials, the governing equation and boundary conditions are written as [27]  
\[\mathbf{\
abla}_{\text{s}}^{\text{T}}\mathbf{\sigma}+\mathbf{b}=\mathbf{0} \tag{1}\]  
\[\mathbf{n}\mathbf{\sigma}=\mathbf{\bar{t}}\;\square\;on\;\mathbf{\Gamma} \tag{2}\]  
where \(\mathbf{\
abla}_{\text{s}}\) is an operator matrix of spatial derivatives expressed as  
\[\mathbf{\
abla}_{\text{s}}^{\text{T}}= \begin{bmatrix}\frac{\partial}{\partial x} &0&0&0&\frac{\partial}{\partial x}&\frac{\partial}{\partial y}\\ 0&\frac{\partial}{\partial y}&0&\frac{\partial}{\partial x}&0&\frac{\partial }{\partial x}\\ 0&0&\frac{\partial}{\partial y}&\frac{\partial}{\partial x}&\frac{\partial}{ \partial x}&0\end{bmatrix}  \tag{3}\]  
\(\mathbf{\sigma}\) is a row vector consisting of components of the stress tensor that is expressed as  
\[\mathbf{\sigma}= \begin{bmatrix}\sigma_{x}&\sigma_{y}&\sigma_{z}&\tau_{yz}&\tau_{ xx}&\tau_{xy}\end{bmatrix} ^{T} \tag{4}\]  
and \(\mathbf{b}\) is body force per unit volume acting on the elastic body. \(\mathbf{\bar{t}}\) is the surface of the body where traction is known as \(\mathbf{\bar{t}}\), and \(\mathbf{n}\) is the direction cosine of the surface. Integrating over the whole domain \(\Omega\), with multiplication by arbitrary trial function \(\mathbf{W}\), a weak form of the governing equation is obtained as follows. \[\int_{\mathbf{\Omega}}\left(\mathbf{\
abla}_{\text{s}}\mathbf{W}\right)^{\text{T}}\!\mathbf{D} \mathbf{\
abla}_{\text{s}}\mathbf{u}\mathbf{d}\mathbf{\Omega}=\int_{\mathbf{\bar{t}}}\mathbf{W}^{\text {T}}\!\mathbf{\bar{t}}\mathbf{d}\mathbf{\Gamma}+\int_{\mathbf{\Omega}}\mathbf{W}^{\text{T}}\!\mathbf{ b}\mathbf{d}\mathbf{\Omega}\;\forall\mathbf{W}^{\text{T}} \tag{5}\]  
The finite element discretization of the equation above is conducted as  
\[\sum_{\mathbf{\epsilon}=1}^{n}\mathbf{W}_{d}^{\text{T}}\!\left(\int_{\mathbf{\Omega}}\mathbf{ B}^{\text{T}}\!\mathbf{D}\mathbf{B}\mathbf{d}\mathbf{\Omega}\mathbf{u}_{d}-\int_{\mathbf{\bar{t}}}\mathbf{N}^{ \text{T}}\!\mathbf{\bar{t}}\mathbf{d}\mathbf{\Gamma}-\int_{\mathbf{\Omega}}\mathbf{N}^{\text{T}}\! \mathbf{b}\mathbf{d}\mathbf{\Omega}\right)=\mathbf{0} \tag{6}\]  
Following relationships for trial function, displacements, and strains are used in the equation above. \[\mathbf{W}=\mathbf{NW}_{\mathbf{d}} \tag{7}\]  
\[\mathbf{u}=\mathbf{Nu}_{\mathbf{d}} \tag{8}\]  
\[\mathbf{\varepsilon}=\mathbf{\
abla}_{\text{s}}\mathbf{Nu}_{\mathbf{d}}=\mathbf{Bu}_{\mathbf{d}} \tag{9}\]  
where \(\mathbf{N}\) is the interpolation function in an element, and \(\mathbf{W}_{\mathbf{d}}\) and \(\mathbf{u}_{\mathbf{d}}\) are nodal trial vectors and nodal displacement vectors, respectively. By letting \(\mathbf{\varepsilon}\) as a row vector consist of components of strain in the same manner as the stress, the relationship between stress and strain for linear elastic materials is defined as  
\[\mathbf{\sigma}=\mathbf{D}\mathbf{\varepsilon} \tag{10}\]  
For dynamic problems, governing Equation (1) can be rewritten in the following form by introducing Corioli's force and viscous damping force into \(\mathbf{b}\)[28]:  
\[\mathbf{\
abla}_{s}^{\text{T}}\mathbf{\sigma}+\left(\mathbf{b}^{\prime}-\mu\frac{d\mathbf{u}} {dt}-\rho\frac{d^{2}\mathbf{u}}{dt^{2}}\right)=\mathbf{0} \tag{11}\]  
where \(\mu\) and \(\rho\) are the viscous damping coefficient and the mass density of the material, respectively. By a similar means of finite element discretization conducted for Equations (6)-(10), the following equation is obtained.