Triangle class for Diffusion

Coordinate-free formulation for a bi-linear diffusion triangle.

Theory

Coordinate mapping:

\[\mathbf{X}=\mathbf{X}_i \xi^i \quad i=1,2,3 \quad \xi^i \geqslant 0 ~\&~ \xi^1+\xi^2+\xi^3 \leq 1\]

Definition of covariant and dual base vectors:

\[d\mathbf{X}=\mathbf{G}_i d \xi^i \rightarrow d \xi^i=\mathbf{G}^i \cdot d \mathbf{X} \quad w/ ~~ \mathbf{G}^i \mathbf{G}_j=\delta_j^i\]

The dual base for the 0-direction of the triangle coordinates:

\[\mathbf{G}^u = -\mathbf{G}^s - \mathbf{G}^t\]

Leads to the following expression for the gradient of the scalar potential function:

\[\nabla \phi = \phi_0 \, \mathbf{G}^u + \phi_1 \, \mathbf{G}^s + \phi_2 \, \mathbf{G}^t\]

Nodal forces

\[\mathbf{F}_I = A t \lambda \, \mathbf{G}^I\cdot\nabla\phi \qquad\text{with}\quad I=u,s,t\]

Nodal tangent stiffness

\[ \left({K}_{t}\right)^{IJ} = A t \lambda \, \mathbf{G}^I\cdot\mathbf{G}^J \qquad\text{with}\quad I,J=u,s,t\]

\[\begin{split}[\mathbf{K}_{t}] = \left[ \begin{array}{ccc} \left({K}_{t}\right)^{uu} & \left({K}_{t}\right)^{us} & \left({K}_{t}\right)^{ut} \\ \left({K}_{t}\right)^{su} & \left({K}_{t}\right)^{ss} & \left({K}_{t}\right)^{st} \\ \left({K}_{t}\right)^{tu} & \left({K}_{t}\right)^{ts} & \left({K}_{t}\right)^{tt} \end{array} \right] = A t \lambda \left[ \begin{array}{ccc} \mathbf{G}^u\cdot\mathbf{G}^u & \mathbf{G}^u\cdot\mathbf{G}^s & \mathbf{G}^u\cdot\mathbf{G}^t \\ \mathbf{G}^s\cdot\mathbf{G}^u & \mathbf{G}^s\cdot\mathbf{G}^s & \mathbf{G}^s\cdot\mathbf{G}^t \\ \mathbf{G}^t\cdot\mathbf{G}^u & \mathbf{G}^t\cdot\mathbf{G}^s & \mathbf{G}^t\cdot\mathbf{G}^t \end{array} \right]\end{split}\]

Parent class

• Element classes

See also

• Triangle6 class for Diffusion

Class doc

class femedu.elements.diffusion.Triangle.Triangle(node0, node1, node2, material)

class: representing a 3-node triangle for diffusion problems

computeSurfaceLoads()

compute surface loads using faces

This method should be called during updateState() by every element supporting surface loads

resetLoads()

default implementation for resetting element loads.

setSurfaceLoad(face, qn)

face ID	nodes defining that face
0	node 0 to node 1
1	node 1 to node 2
2	node 2 to node 0

Parameters:

• face – face ID for the loaded face

• qn – magnitude of distributed heat flux \(q_n\). Influx on a surface is positive.

updateState()