The linearization and discretization of the formulation with surface integrals given in Sect.
3.1 is considered. This yields force type contributions (
\({\varvec{f}}_{1}\) to
\({\varvec{f}}_{8}\)), which are subtracted from the external applied forces
\({\varvec{f}}_{ext}\), and tangent type contributions (
\(\varvec{\underline{K}}_{1}\) to
\(\varvec{\underline{K}}_{8}\)), which aid in the search for the increment of the state variable
\(\varvec{\varphi }\) in a
Newton-
Raphson iterative solution scheme [
12]. Since the transient case is considered, there are terms that depend on the velocity and acceleration in the ALE reference frame. Hence, the
Newton-
Raphson iterative solution scheme for values at iteration
\(j+1\) based on known quantities at iteration
j, can be expressed as
$$\begin{aligned} \begin{aligned}&\varvec{\underline{M}}^{j}\cdot \Delta \frac{\partial ^{2}\varvec{\varphi }}{\partial t^{2}} + \varvec{\underline{D}}^{j}\cdot \Delta \frac{\partial \varvec{\varphi }}{\partial t} + \varvec{\underline{G}}^{j}\cdot \Delta \varvec{\varphi } = {\varvec{f}}_{ext}^{j+1} - {\varvec{f}}_{int}^{j} \;. \end{aligned}\nonumber \\ \end{aligned}$$
(14)
The above Eq. (
14) can be expressed in terms of the individual tangent contributions (
\(\varvec{\underline{K}}_{1}\) to
\(\varvec{\underline{K}}_{8}\)), and internal force contributions (
\({\varvec{f}}_{1}\) to
\({\varvec{f}}_{8}\)) as
$$\begin{aligned} \begin{aligned}&\varvec{\underline{K}}_{2}^{j}\cdot \Delta \frac{\partial ^{2}\varvec{\varphi }}{\partial t^{2}} + \left( \varvec{\underline{K}}_{3}^{j}+\varvec{\underline{K}}_{5}^{j}+\varvec{\underline{K}}_{7}^{j}\right) \cdot \Delta \frac{\partial \varvec{\varphi }}{\partial t} \\&\quad \left( \varvec{\underline{K}}_{4}^{j}+\varvec{\underline{K}}_{6}^{j}+\varvec{\underline{K}}_{8}^{j}\right) \cdot \Delta \varvec{\varphi } = {\varvec{f}}_{ext}^{j+1} - \sum _{i=1}^{8}{\varvec{f}}_{i}^{j} \;, \end{aligned} \end{aligned}$$
(15)
where it is to be noted, that since the first term of Eq. (
12) does not contain any terms that depend on the state variable
\(\varvec{\varphi }\). Hence, there is no associated tangent contribution (
\(\varvec{\underline{K}}_{1} = \varvec{\underline{0}}\)). The linearization of each term in Eq. (
12) is described separately for the sake of clarity. The first term can be expressed in terms of the rate of change of the guiding velocity
\({\varvec{w}}\) as
$$\begin{aligned} \int _{\chi }{\hat{\rho }}\frac{\partial ^{2}\varvec{\chi }_\text {0}^\text {ALE}}{\partial t^{2}}\Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} = -\int _{\chi }{\hat{\rho }}\frac{\partial {\varvec{w}}}{\partial t}\Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} \;. \end{aligned}$$
(16)
The rate of change of guiding velocity can be calculated from the prescribed values of guiding velocity at each time step, and is known explicitly at each time step. Therefore, there is only a force type contribution from the first term. This can be described, after discretization for an arbitrary finite element
e as
$$\begin{aligned} \widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{1}^{e} = -\widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial {\varvec{w}}}}{\partial t}\Bigg \vert _{\chi }d\hat{V}^{e} \;, \end{aligned}$$
(17)
where the term
\(\varvec{\underline{H}}\) is a matrix with orderly arrangement of the shape functions, and
\(\varvec{\eta }\) is an arbitrary test function. The matrix
\(\varvec{\underline{H}}\) can be written as
$$\begin{aligned} \begin{aligned} \varvec{\underline{H}}(\varvec{\xi }) = \left[ \begin{array}{ccccccc} N_{1}(\varvec{\xi }) &{} 0 &{} 0 &{} \cdots &{} N_{k}(\varvec{\xi }) &{} 0 &{} 0 \\ 0 &{} N_{1}(\varvec{\xi }) &{} 0 &{} \cdots &{} 0 &{} N_{k}(\varvec{\xi }) &{} 0 \\ 0 &{} 0 &{} N_{1}(\varvec{\xi }) &{} \cdots &{} 0 &{} 0 &{} N_{k}(\varvec{\xi }) \end{array}\right] , \end{aligned}\nonumber \\ \end{aligned}$$
(18)
where
\(N_{k}(\varvec{\xi })\) refers to the shape function associated with iso-parametric coordinates
\(\varvec{\xi }\) and node
k of a given finite element. Further, the matrix
\(\varvec{\underline{H}}\) is used to project the nodal values of any quantity
\({\widetilde{\square }}\) to the integration points in the finite element (e.g.
\(\varvec{\eta }(\varvec{\xi }) = \varvec{\underline{H}}\cdot \widetilde{\varvec{\eta }}\)). Moreover, the rate of change of the guiding velocity is constant throughout the finite element mesh, and so, the above Eq. (
17) can be written as
$$\begin{aligned} \widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{1}^{e} = -\int _{\chi ^{e}}{\hat{\rho }}\cdot \varvec{\underline{H}}^{T}\cdot \frac{\partial {\varvec{w}}}{\partial t}\Bigg \vert _{\chi }d\hat{V}^{e} \;. \end{aligned}$$
(19)
The second term depends on the acceleration in the ALE frame, and can be linearized with respect to the acceleration as
$$\begin{aligned} \begin{aligned}&\int _{\chi }{\hat{\rho }}\left( \frac{\partial ^{2}\varvec{\varphi }}{\partial t^{2}}\Bigg \vert _{\chi }\right) _{i+1}\cdot \varvec{\eta }d\hat{V} = \int _{\chi }{\hat{\rho }}\left( \frac{\partial ^{2}\varvec{\varphi }}{\partial t^{2}}\Bigg \vert _{\chi }\right) _{i}\cdot \varvec{\eta }d\hat{V}\\&\quad + \int _{\chi }{\hat{\rho }}\left( \Delta \frac{\partial ^{2}\varvec{\varphi }}{\partial t^{2}}\right) \Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} \;. \end{aligned} \end{aligned}$$
(20)
The discretization of the above Eq. (
20) results in contributions to both the forces and the tangent given by
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{2}^{e} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial ^{2}\varvec{\varphi }}}{\partial t^{2}}\Bigg \vert _{\chi }d\hat{V}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{2}^{e}\cdot \Delta \frac{\widetilde{\partial ^{2}\varvec{\varphi }}}{\partial t^{2}} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}} \; d\hat{V}^{e}\cdot \Delta \frac{\widetilde{\partial ^{2}\varvec{\varphi }}}{\partial t^{2}} \;, \end{aligned} \end{aligned}$$
(21)
respectively. The third term in Eq. (
12) depends on the velocity in the ALE frame, and can be linearized with respect to the velocity as
$$\begin{aligned} \begin{aligned}&\int _{\chi }{\hat{\rho }}\left[ \text {Grad}\left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i+1}\cdot {\varvec{w}}\right] \cdot \varvec{\eta }d\hat{V} \\&\quad =\int _{\chi }{\hat{\rho }}\left[ \text {Grad}\left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i}\cdot {\varvec{w}}\right] \cdot \varvec{\eta }d\hat{V} \\&\qquad +\int _{\chi }{\hat{\rho }}\left[ \text {Grad}\Delta \frac{\partial \varvec{\varphi }}{\partial t}\cdot {\varvec{w}}\right] \cdot \varvec{\eta }d\hat{V} \;. \end{aligned} \end{aligned}$$
(22)
The discretization of the above Eq. (
22) results in contributions to both the forces and the tangent given by
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{3}^{e} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Bigg \vert _{\chi }d\hat{V}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{3}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}} \; d\hat{V}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} \;, \end{aligned} \end{aligned}$$
(23)
respectively. Here, the term
\(\varvec{\underline{A}}\) refers to the following matrix
$$\begin{aligned} \begin{aligned}&\varvec{\underline{A}} = \left[ \begin{array}{ccccccc} N_{1,i}w_{i} &{} 0 &{} 0 &{} \tiny {\cdots } &{} N_{k,i}w_{i} &{} 0 &{} 0 \\ 0 &{} N_{1,i}w_{i} &{} 0 &{} \tiny {\cdots } &{} 0 &{} N_{k,i}w_{i} &{} 0 \\ 0 &{} 0 &{} N_{1,i}w_{i} &{} \tiny {\cdots } &{} 0 &{} 0 &{} N_{k,i}w_{i} \end{array}\right] \;, \end{aligned} \end{aligned}$$
(24)
where
\(N_{k,i}w_{i}\) is defined as the summation
\(\sum _{i}N_{k,i}w_{i}\).
\(N_{k,i}\) is the derivative of the shape function associated with node
k along the
i direction, and
\(w_{i}\) is the component of the guiding velocity along the
i direction. The fourth term in Eq. (
12) depends on the current coordinates within the ALE frame, and can be linearized with respect to this as
$$\begin{aligned} \begin{aligned}&\int _{\chi }{\hat{\rho }}\;\text {Grad}\left( \varvec{\varphi }\right) _{i+1}\cdot \frac{\partial {\varvec{w}}}{\partial t}\Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} = \int _{\chi }{\hat{\rho }}\;\text {Grad}\left( \varvec{\varphi }\right) _{i}\cdot \frac{\partial {\varvec{w}}}{\partial t}\Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} \\&\quad + \int _{\chi }{\hat{\rho }}\;\text {Grad}\left( \Delta \varvec{\varphi }\right) \cdot \frac{\partial {\varvec{w}}}{\partial t}\Bigg \vert _{\chi }\cdot \varvec{\eta }d\hat{V} \;. \end{aligned} \end{aligned}$$
(25)
The discretization of the above Eq. (
25) results in contributions to both the forces and the tangent, expressed as
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{4}^{e} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A'}}\cdot \widetilde{\varvec{\varphi }}d\hat{V}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{4}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A'}} \; d\hat{V}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} \;, \end{aligned} \end{aligned}$$
(26)
respectively. Here, the matrix
\(\varvec{\underline{A'}}\) is defined as
$$\begin{aligned} \begin{aligned} \varvec{\underline{A}}' = \left[ \begin{array}{ccccccc} N_{1,i}{\dot{w}}_{i} &{} 0 &{} 0 &{} \tiny {\cdots } &{} N_{k,i}{\dot{w}}_{i} &{} 0 &{} 0 \\ 0 &{} N_{1,i}{\dot{w}}_{i} &{} 0 &{} \tiny {\cdots } &{} 0 &{} N_{k,i}{\dot{w}}_{i} &{} 0 \\ 0 &{} 0 &{} N_{1,i}{\dot{w}}_{i} &{} \tiny {\cdots } &{} 0 &{} 0 &{} N_{k,i}{\dot{w}}_{i} \end{array}\right] \;, \end{aligned}\nonumber \\ \end{aligned}$$
(27)
where
\({\dot{w}}_{i}\) refers to the component of
\(\frac{\partial {\varvec{w}}}{\partial t}\big \vert _{\chi }\) along the
i direction. It is important to note that the matrix
\(\varvec{\underline{A'}}\) can be determined in a relatively straightforward fashion in the case of pavements, since all nodes in the mesh have the same guiding velocity at any given point of time. However, in the case of rolling wheels, since there is generally a non-zero angular guiding velocity, the determination of this matrix requires the storage of time history data of the guiding velocity itself. The fifth term in Eq. (
12) can be linearized with respect to the velocity in the ALE reference frame as
$$\begin{aligned} \begin{aligned}&-\int _{\chi }{\hat{\rho }}\left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i+1}\cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V}\\&\quad = -\int _{\chi }{\hat{\rho }}\left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i}\cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V} \\&\qquad -\int _{\chi }{\hat{\rho }}\Delta \frac{\partial \varvec{\varphi }}{\partial t}\cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V} \;. \end{aligned} \end{aligned}$$
(28)
The discretization of the above Eq. (
28) results in terms that contribute to both forces and the tangent, and can be described by
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{5}^{e} = -\widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Bigg \vert _{\chi }d\hat{V}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{5}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} = -\widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{H}}\;d\hat{V}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} \;, \end{aligned} \end{aligned}$$
(29)
respectively. The sixth term of Eq. (
12) can be linearized with respect to the current coordinates in the ALE reference frame, and this results in
$$\begin{aligned} \begin{aligned}&-\int _{\chi }{\hat{\rho }}\left( \text {Grad}\left( \varvec{\varphi }\right) _{i+1}\cdot {\varvec{w}}\right) \cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V} \\&\quad =-\int _{\chi }{\hat{\rho }}\left( \text {Grad}\left( \varvec{\varphi }\right) _{i}\cdot {\varvec{w}}\right) \cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V} \\&\qquad -\int _{\chi }{\hat{\rho }}\left( \text {Grad}\Delta \varvec{\varphi }\cdot {\varvec{w}}\right) \cdot \left( \text {Grad}\varvec{\eta }\cdot {\varvec{w}}\right) d\hat{V} \;. \end{aligned} \end{aligned}$$
(30)
Equation (
30) can be discretized and yields contributions to both forces and the tangent. These force and tangent contributions are described as
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{6}^{e} = -\widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{A}}\cdot \widetilde{\varvec{\varphi }}d\hat{V}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{6}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} = -\widetilde{\varvec{\eta }}^{T}\cdot \int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{A}} \; d\hat{V}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} \;, \end{aligned} \end{aligned}$$
(31)
respectively. The seventh and eighth terms in Eq. (
12) are surface integrals and need to be treated for each surface of every finite element. The seventh term in Eq. (
12) depends on the velocity in the ALE reference frame, and can be linearized as
$$\begin{aligned} \begin{aligned}&\int _{\partial \chi }{\hat{\rho }}\varvec{\eta }\cdot \left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i+1}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \\&\quad =\int _{\partial \chi }{\hat{\rho }}\varvec{\eta }\cdot \left( \frac{\partial \varvec{\varphi }}{\partial t}\Bigg \vert _{\chi }\right) _{i}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \\&\qquad +\int _{\partial \chi }{\hat{\rho }}\varvec{\eta }\cdot \Delta \frac{\partial \varvec{\varphi }}{\partial t}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \;. \end{aligned} \end{aligned}$$
(32)
Equation (
32) contributes to both force and tangent terms when discretized as demonstrated below
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{7}^{e} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Bigg \vert _{\chi }\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{7}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\cdot \Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t} \;, \end{aligned}\nonumber \\ \end{aligned}$$
(33)
respectively. Finally, the eighth term in Eq. (
12) can be linearized with respect to the current coordinates in the ALE reference frame, as
$$\begin{aligned} \begin{aligned}&\int _{\partial \chi }{\hat{\rho }}\left( \varvec{\eta }\cdot \left( \text {Grad}\left( \varvec{\varphi }\right) _{i+1}\cdot {\varvec{w}}\right) \right) \left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \\&\quad = \int _{\partial \chi }{\hat{\rho }}\left( \varvec{\eta }\cdot \left( \text {Grad}\left( \varvec{\varphi }\right) _{i}\cdot {\varvec{w}}\right) \right) \left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \\&\qquad +\int _{\partial \chi }{\hat{\rho }}\left( \varvec{\eta }\cdot \left( \text {Grad}\Delta \varvec{\varphi }\cdot {\varvec{w}}\right) \right) \left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A} \;. \end{aligned} \end{aligned}$$
(34)
The above Eq. (
34) can then be discretized and contributes to both forces and the tangent. These are given by
$$\begin{aligned} \begin{aligned}&\widetilde{\varvec{\eta }}^{T}\cdot {\varvec{f}}_{8}^{e} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\cdot \widetilde{\varvec{\varphi }}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e} \;, \\&\text {and} \\&\widetilde{\varvec{\eta }}^{T}\cdot \varvec{\underline{K}}_{8}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} = \widetilde{\varvec{\eta }}^{T}\cdot \int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\cdot \Delta \widetilde{\varvec{\varphi }} \;, \end{aligned} \end{aligned}$$
(35)
respectively. Thus, all the terms of Eq. (
12) are discretized and can be implemented in a finite element framework. All the element tangent type contributions are assembled into the global material tangent, and all the element forces are assembled into the global internal forces array. The summary of the discretized terms from Eq. (
12) are provided in Table
1, after discarding the arbitrary test function
\(\varvec{\eta }\).
Table 1
Summary of the discretized terms obtained by expanding out the inertia term of the balance of linear momentum considering surface terms
1 | \(\varvec{\underline{0}}\) | – | \(-\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \frac{\partial {\varvec{w}}}{\partial t}\Big \vert _{\chi }d\hat{V}^{e}\) |
2 | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}} d\hat{V}^{e}\) | \(\Delta \frac{\widetilde{\partial ^{2}\varvec{\varphi }}}{\partial t^{2}}\) | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial ^{2}\varvec{\varphi }}}{\partial t^{2}}\Big \vert _{\chi }d\hat{V}^{e}\) |
3 | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}} d\hat{V}^{e}\) | \(\Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\) | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Big \vert _{\chi }d\hat{V}^{e}\) |
4 | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A'}}d\hat{V}^{e}\) | \(\Delta \widetilde{\varvec{\varphi }}\) | \(\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A'}}\cdot \widetilde{\varvec{\varphi }}d\hat{V}^{e}\) |
5 | \(-\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{H}}d\hat{V}^{e}\) | \(\Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\) | \(-\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Big \vert _{\chi }d\hat{V}^{e}\) |
6 | \(-\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{A}}d\hat{V}^{e}\) | \(\Delta \varvec{{\widetilde{\varphi }}}\) | \(-\int _{\chi ^{e}}{\hat{\rho }}\varvec{\underline{A}}^{T}\cdot \varvec{\underline{A}}\cdot \widetilde{\varvec{\varphi }}d\hat{V}^{e}\) |
7 | \(\int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\) | \(\Delta \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\) | \(\int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{H}}\cdot \frac{\widetilde{\partial \varvec{\varphi }}}{\partial t}\Big \vert _{\chi }\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\) |
8 | \(\int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\) | \(\Delta \widetilde{\varvec{\varphi }}\) | \(\int _{\partial \chi ^{e}}{\hat{\rho }}\varvec{\underline{H}}^{T}\cdot \varvec{\underline{A}}\cdot \widetilde{\varvec{\varphi }}\left( {\varvec{w}}\cdot \varvec{\hat{N}}\right) d\hat{A}^{e}\) |