Introduction

Theory

$$M(I,J)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}\frac{\partial N^{J}T_{b}}{\partial x_{i}}d\Omega dt$$ $$M(I,J)=\int_{I_{n}}\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}\frac{\partial N^{J}T_{b}}{\partial x_{i}}d\Omega dt$$ $$M(I,J)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}k_{ij}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$ $$M(I,J)=\int_{I_{n}}\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}k_{ij}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$ $$M(I,J)=\int_{I_{n}}\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}a_{i}b_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$ $$M(I,J)=\int_{I_{n}}\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}a_{i}b_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt$$

Methods

STDiffusionMatrix 1

General interface is given below.

mat4 = STDiffusionMatrix(test, trial)
mat2 = DiffusionMatrix(test, trial)

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

STDiffusionMatrix 2

General interface is given below.

mat4 = STDiffusionMatrix(test, trial, k)
mat2 = DiffusionMatrix(test, trial, k)

!!! example ""

$$M(I,J)=\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}\frac{\partial N^{J}T_{b}}{\partial x_{i}}d\Omega$$

Here, $\rho$ is a scalar [[FEVariable_]], therefore, we can obtain this matrix by following call.

mat4 = STDiffusionMatrix(test, trial, k=rho)
mat2 = DiffusionMatrix(test, trial, k=rho)

!!! example ""

$$M(I,J)=\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}k_{ij}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega$$

Here, $k_{ij}$ is a diffusion tensor [[FEVariable_]], therefore, we can obtain this matrix by following call.

mat4 = STDiffusionMatrix(test, trial, k=k)
mat2 = DiffusionMatrix(test, trial, k=k)

STDiffusionMatrix 3

The general interface in this category is given below.

mat4 = STDiffusionMatrix(test, trial, c1, c2)
mat2 = DiffusionMatrix(test, trial, c1, c2)

!!! note "c1" c1 is a [[FEVariable_]], which can be a scalar, vector, or a matrix.

!!! note "c2" c2 is a [[FEVariable_]], which can be a scalar, vector, or a matrix.

!!! note "" 9 types of matrices can be formed from this routine.

!!! example "scalar-scalar"

!!! example "scalar-vector"

!!! example "scalar-matrix"

$$M(I,J)=\int_{\Omega}\rho\frac{\partial N^{I}T_{a}}{\partial x_{i}}k_{ij}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega$$

Here $\rho$ is a scalar and $k$ is a matrix (rank-2 tensor) [[FEVariable_]]. This matrix can be obtained by following expression.

mat4 = STDiffusionMatrix(test, trial, c1=rho, c2=k)
mat2 = DiffusionMatrix(test, trial, c1=rho, c2=k)

STDiffusionMatrix 4

Generic interface is given below.

mat4 = STDiffusionMatrix(test, trial, c1, c2, c3)
mat2 = DiffusionMatrix(test, trial, c1, c2, c3)

$$M(I,J)=\int_{\Omega}\frac{\partial N^{I}T_{a}}{\partial x_{i}}a_{i}b_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega$$