STConvectiveMatrix

Theory

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial u_i}{\partial t} + c_k \frac{\partial u_i}{\partial x_k} + \cdots$$

We would like to compute the following matrices.

$${}^{4}M(I,J,a,b) = {}^{a}\delta u_{iI} \quad \int_{Q_n} N^I T_a c_k \frac{\partial N^J T_b}{\partial x_k} {dQ} \quad {}^{b}u_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta u_{iI} \quad \int_{Q_n} c_k \frac{\partial N^I T_a}{\partial x_k} N^J T_b {dQ} \quad {}^{b}u_{iJ}$$

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial \textbf{U}}{\partial t} + \frac{\partial \textbf{f(U)}}{\partial x} + \frac{\partial \textbf{g(U)}}{\partial y} + \frac{\partial \textbf{h(U)}}{\partial z} + \cdots$$

where $\textbf{U}, \textbf{f}, \textbf{g}, \textbf{h} \in R^m$. In this case we wish to compute the following matrices.

$${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} N^I T_a \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}f_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial x} N^J T_b {dQ} \quad {}^{b}f_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} N^I T_a \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}g_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial x} N^J T_b {dQ} \quad {}^{b}g_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} N^I T_a \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}h_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a} U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial x} N^J T_b {dQ} \quad {}^{b}h_{iJ}$$

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial u_i}{\partial t} + c_k \frac{\partial u_i}{\partial x_k} + \cdots$$

We would like to compute the following matrices.

$${}^{4}M(I,J,a,b) = \delta {}^{a}u_{iI} \int_{Q_n} \frac{ \partial N^I T_a}{\partial t} c_{k}^{h} \frac{\partial N^J T_b}{\partial x_k} {dQ} \quad {}^{b}u_{iJ}$$ $${}^{4}M(I,J,a,b) = \delta {}^{a}u_{iI} \int_{Q_n} c_{k}^{h} \frac{\partial N^I T_a}{\partial x_k} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}u_{iJ}$$

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial \textbf{U}}{\partial t} + \frac{\partial \textbf{f(U)}}{\partial x} + \frac{\partial \textbf{g(U)}}{\partial y} + \frac{\partial \textbf{h(U)}}{\partial z} + \cdots$$

where $\textbf{U}, \textbf{f}, \textbf{g}, \textbf{h} \in R^m$. In this case we wish to compute the following matrices.

$${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}f_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial x} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}f_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} \frac{\partial N^J T_b}{\partial y} {dQ} \quad {}^{b}g_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial y} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}g_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} \frac{\partial N^J T_b}{\partial z} {dQ} \quad {}^{b}h_{iJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial z} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}h_{iJ}$$

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial \textbf{U}}{\partial t} + \mathbf{A_1} \frac{\partial \textbf{U}}{\partial x} + \mathbf{A_2} \frac{\partial \textbf{U}}{\partial y} + \mathbf{A_3} \frac{\partial \textbf{U}}{\partial z} + \cdots$$

where $\textbf{U} \in R^m$, $\mathbf{A_i} \in R^{m \times m}$. In this case we wish to compute the following matrices.

$${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} N^I T_a [ \mathbf{A_1} ]_{ij} \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_1} ]_{ji} \frac{\partial N^I T_a}{\partial x} N^J T_b {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} N^I T_a [\mathbf{A_2}]_{ij} \frac{\partial N^J T_b}{\partial y} {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_2}]_{ji} \frac{\partial N^I T_a}{\partial y} N^J T_b {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} N^I T_a [\mathbf{A_3}]_{ij} \frac{\partial N^J T_b}{\partial z} {dQ} \quad {}^{b}h_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_3}]_{ji} \frac{\partial N^I T_a}{\partial z} N^J T_b {dQ} \quad {}^{b}h_{jJ}$$

The shape of each ${}^{4}M(:,:,a,b)$ is $(N_{NS} \times m, N_{NS} \times m)$. In this case there will be coupling between different components of $\mathbf{U}$. This coupling is due to $\mathbf{A_i}$. The structure of any of the above ${}^{4}\mathbf{M}$ is given as

$${}^{4}\mathbf{M}(:,:,a,b) = \begin{bmatrix} \mathbf{M_{11}} & \cdots & \mathbf{M_{1m}} \\ \vdots & \ddots & \vdots \\ \mathbf{M_{m1}} & \cdots & \mathbf{M_{mm}} \\ \end{bmatrix}$$

Each $\mathbf{M_{ij}}$ has shape $(N_{ns} \times N_{ns})$.

Now we want to compute the space-time convective finite element matrix for following PDE.

$$\frac{\partial \textbf{U}}{\partial t} + \mathbf{A_1} \frac{\partial \textbf{U}}{\partial x} + \mathbf{A_2} \frac{\partial \textbf{U}}{\partial y} + \mathbf{A_3} \frac{\partial \textbf{U}}{\partial z} + \cdots$$

where $\textbf{U} \in R^m$, $\mathbf{A_i} \in R^{m \times m}$. In this case we wish to compute the following matrices.

$${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} [ \mathbf{A_1} ]_{ij} \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_1} ]_{ji} \frac{\partial N^I T_a}{\partial x} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} [\mathbf{A_2}]_{ij} \frac{\partial N^J T_b}{\partial y} {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_2}]_{ji} \frac{\partial N^I T_a}{\partial y} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} \frac{\partial N^I T_a}{\partial t} [\mathbf{A_3}]_{ij} \frac{\partial N^J T_b}{\partial z} {dQ} \quad {}^{b}h_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_3}]_{ji} \frac{\partial N^I T_a}{\partial z} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}h_{jJ}$$

The shape of each ${}^{4}M(:,:,a,b)$ is $(N_{NS} \times m, N_{NS} \times m)$. In this case there will be coupling between different components of $\mathbf{U}$. This coupling is due to $\mathbf{A_i}$. The structure of any of the above ${}^{4}\mathbf{M}$ is given as

$${}^{4}\mathbf{M}(:,:,a,b) = \begin{bmatrix} \mathbf{M_{11}} & \cdots & \mathbf{M_{1m}} \\ \vdots & \ddots & \vdots \\ \mathbf{M_{m1}} & \cdots & \mathbf{M_{mm}} \\ \end{bmatrix}$$

Each $\mathbf{M_{ij}}$ has shape $(N_{ns} \times N_{ns})$.

Now consider the following terms in a pde.

$$\mathbf{A_0} \frac{\partial U}{\partial t} + \mathbf{A_1} \frac{\partial U}{\partial x} + \mathbf{A_2} \frac{\partial \mathbf{U}}{\partial y} + \mathbf{A_3} \frac{\partial U}{\partial t} + \cdots$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_0}]_{ki} \frac{\partial N^I T_a}{\partial t} [ \mathbf{A_1} ]_{kj} \frac{\partial N^J T_b}{\partial x} {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_1} ]_{ki} \frac{\partial N^I T_a}{\partial x} [\mathbf{A_0}]_{kj} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}f_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_0}]_{ki}\frac{\partial N^I T_a}{\partial t} [\mathbf{A_2}]_{kj} \frac{\partial N^J T_b}{\partial y} {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_2}]_{ki} \frac{\partial N^I T_a}{\partial y} [ \mathbf{A_0}]_{kj} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}g_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [ \mathbf{A_0}]_{ki}\frac{\partial N^I T_a}{\partial t} [\mathbf{A_3}]_{kj} \frac{\partial N^J T_b}{\partial z} {dQ} \quad {}^{b}h_{jJ}$$ $${}^{4}M(I,J,a,b) = {}^{a}\delta U_{iI} \quad \int_{Q_n} [\mathbf{A_3}]_{ki} \frac{\partial N^I T_a}{\partial z} [ \mathbf{A_0}]_{kj} \frac{\partial N^J T_b}{\partial t} {dQ} \quad {}^{b}h_{jJ}$$

Methods

STConvectiveMatrix 1

Implementation: STCM_1

!!! note "Interface"

CALL ConvectiveMatrix( test, trial, c, term1, term2)

!!! example ""

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{c_j}\frac{{\partial {N^I}{T_a}}}{{\partial {x_j}}} \cdot {N^J}{T_b}d\Omega dt} } }$$

CALL ConvectiveMatrix(test, trial, c, term1=del_x, term2=del_none, projectOn='test')

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}N^{I}T_{a}c_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$

CALL ConvectiveMatrix(test, trial, c, term1=del_none, term2=del_x, projectOn='trial')

• C denotes the convective velocity. It can be Rank-1, Rank-2, Rank-3 array wrapped inside the [[FEVariable_]].
• If the convective velocity is constant in both space and time, C should be a rank-1, C(:).
• The spatial nodal values of convective velocity, when it is constant in time domain, is given by a rank-2 array C(:,:)
• The temporal values of convective velocity, when it is constant in time domain, is given by a rank-2 array C(:,:)
• Space-time nodal values of convective velocity is given by a rank-3 array C(:,:,:)

You can learn more about this method from following pages

• [[STConvectiveMatrix_test_1]] Line2 for space and time, constant velocity
• [[STConvectiveMatrix_test_2]] Line2 for space and time, spatial nodal velocity
• [[STConvectiveMatrix_test_3]] Line2 for space and time, space-time nodal velocity
• [[STConvectiveMatrix_test_4]] Line2 for space and time, constant velocity

STConvectiveMatrix 2

Implementation: STCM_2 and STCM_3

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {\frac{{\partial {N^I}{T_a}}}{{\partial x}} c \cdot {N^J}{T_b}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_none, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {\frac{{\partial {N^I}{T_a}}}{{\partial y}} c \cdot {N^J}{T_b}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_none, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {\frac{{\partial {N^I}{T_a}}}{{\partial z}} c \cdot {N^J}{T_b}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_z, term2=del_none, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{N^J}{T_b} c \cdot \frac{{\partial {N^J}{T_b}}}{{\partial x}}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_x, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{N^J}{T_b} c \cdot \frac{{\partial {N^J}{T_b}}}{{\partial y}}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_y, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{N^J}{T_b} c \cdot \frac{{\partial {N^J}{T_b}}}{{\partial z}}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_z, c )

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {\frac{{\partial {N^I}{T_a}}}{{\partial x_{i}}} c \cdot {N^J}{T_b}d\Omega dt} } }$$

This matrix can be computed by setting dim=-1 as shown in the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_none,c)

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{N^J}{T_b} c \cdot \frac{{\partial {N^J}{T_b}}}{{\partial x_{i}}}d\Omega dt} } }$$

This matrix can be computed using the following command.

mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_x_all,c)

!!! note "" You can learn more about this subroutine from following pages.

• [[STConvectiveMatrix_test_21]] Line2 in space and time, FEVariable is constant
• [[STConvectiveMatrix_test_22]] Line2 in space and time, FEVariable is space-nodal values
• [[STConvectiveMatrix_test_23]] Line2 in space and time, FEVariable is space-time values
• [[STConvectiveMatrix_test_24]] Triangle3 in space and Line2 in time, FEVariable is constant
• [[STConvectiveMatrix_test_25]] Triangle3 in space and Line2 in time, FEVariable is spatial nodal values
• [[STConvectiveMatrix_test_26]] Triangle3 in space and Line2 in time, FEVariable is space-time nodal values
• [[STConvectiveMatrix_test_27]] Triangle3 in space and Line2 in time, FEVariable is space-time nodal values

STConvectiveMatrix 3

Implementation: STCM_4 and STCM_5.

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial x}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_t,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial y}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_t,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial z}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_z, tzerm2=del_t,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial x}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial y}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_y,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial z}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_z,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial x_{i}}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_t,c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial x_{i}}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x_all,c)

You can learn more about these methods from following pages 🚀💎🔗

• [[STConvectiveMatrix_test_31]]
• [[STConvectiveMatrix_test_32]]

STConvectiveMatrix 4

Implementation: STCM_6 and STCM_7.

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_x,c=c, projectOn="trial")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial y}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_x,c=c, projectOn="trial")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial z}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_z, term2=del_x,c=c, projectOn="trial")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{p}}c_{p}\frac{\partial N^{J}T_{b}}{\partial x}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_x,c=c, projectOn="test")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{p}}c_{p}\frac{\partial N^{J}T_{b}}{\partial y}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_y,c=c, projectOn="test")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{p}}c_{p}\frac{\partial N^{J}T_{b}}{\partial z}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_z,c=c, projectOn="test")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_x,c=c, &
    & projectOn="trial")

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{p}}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_i}d\Omega dt$$

mat2=ConvectiveMatrix( test, trial, term1=del_x, &
    & term2=del_x_all,c=c, projectOn="test")

You can learn more about these methods from following pages.

• [[STConvectiveMatrix_test_41]]
• [[STConvectiveMatrix_test_42]]

STConvectiveMatrix 5

Implementation: STCM_8

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial x_{p}}c_{p}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_y, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_z, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x_all, c)

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{p}}c_{p}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_t, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_t, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_z, term2=del_t, c)
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_t, c)
    !! or

You can learn how to use these methods from following pages.

• [[STConvectiveMatrix_test_51]]
• [[STConvectiveMatrix_test_52]]

STConvectiveMatrix 6

Implementation: STCM_1b

$$M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} \rho {{c_j}\frac{{\partial {N^I}{T_a}}}{{\partial {x_j}}} \cdot {N^J}{T_b}d\Omega dt} } }$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_none, &
        & c=c, rho=rho, projectionOn='test' )
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_none, &
        & c=c, rho=rho, projectionOn='test' )
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_none, &
        & c=c, rho=rho, projectionOn='test' )
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_z, term2=del_none, &
        & c=c, rho=rho, projectionOn='test' )

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega} \rho N^{I}T_{a}c_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_x_all, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_x, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_y, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_none, term2=del_z, &
        & c=c, rho=rho, projectionOn='trial')

You can learn more about this method from following pages

• [[STConvectiveMatrix_test_61]]

STConvectiveMatrix 7

Implementation: STCM_6b and STCM_7b

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho c_{p}\frac{\partial N^{I}T_{a}}{\partial x_{p}}\frac{\partial N^{J}T_{b}}{\partial x} d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_x, &
        & c=c, rho=rho, projectionOn='test')

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho c_{p}\frac{\partial N^{I}T_{a}}{\partial x_{p}}\frac{\partial N^{J}T_{b}}{\partial y}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_y, &
        & c=c, rho=rho, projectionOn='test')

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho c_{p}\frac{\partial N^{I}T_{a}}{\partial x_{p}}\frac{\partial N^{J}T_{b}}{\partial z}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_z, &
        & c=c, rho=rho, projectionOn='test')

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho \frac{\partial N^{I}T_{a}}{\partial x}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_x, &
        & c=c, rho=rho, projectionOn='trial')

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho \frac{\partial N^{I}T_{a}}{\partial y}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_y, &
        & c=c, rho=rho, projectionOn='trial')

$$M(I,J,a,b)=\int_{I_{n}}\int_{\Omega} \rho \frac{\partial N^{I}T_{a}}{\partial z}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_z, &
        & c=c, rho=rho, projectionOn='trial')

You can learn more about this method from following pages.

• [[STConvectiveMatrix_test_71]]

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_x, &
        & c=c, rho=rho, projectionOn='trial')

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}\rho c_{p}\frac{\partial N^{I}T_{a}}{\partial x_{p}}\frac{\partial N^{J}T_{b}}{\partial x_{i}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_x_all, &
        & c=c, rho=rho, projectionOn='test')

You can learn more about this method from following pages

• [[STConvectiveMatrix_test_72]]

STConvectiveMatrix 8

Implementation: STCM_8b

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}\rho c_{p}\frac{\partial N^{I}T_{a}}{\partial x_{p}}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_x, term2=del_t, &
        & c=c, rho=rho, projectionOn='test')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_y, term2=del_t, &
        & c=c, rho=rho, projectionOn='test')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_z, term2=del_t, &
        & c=c, rho=rho, projectionOn='test')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_x_all, term2=del_t, &
        & c=c, rho=rho, projectionOn='test')

$$M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial t}c_{p}\frac{\partial N^{J}T_{b}}{\partial x_{p}}d\Omega dt$$

    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_y, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_z, &
        & c=c, rho=rho, projectionOn='trial')
    !! or
    mat2=ConvectiveMatrix( test, trial, term1=del_t, term2=del_x_all, &
        & c=c, rho=rho, projectionOn='trial')

You can learn more about this method from following pages

• [[STConvectiveMatrix_test_81]]