# traction boundary conditions in elasticity

I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My question is: How is the node order/connectivity for the traction elements defined? In 2D the boundary term $$({\bf{w}},{\bf{h}})_{\Gamma_h}$$ is a line integral. Since the connectivity for elements 3 and 4 is (5,6,9,8) and (4,5,8,7) should the nodes for the traction elements be chosen in the same order? That is, should the four traction elements have the connectivity (6,9), (9,8), (8,7) and (7,4)? That means the boundary integral should be $$\int_{node 6}^{node 9} {\bf{w}}\cdot{\bf{h}} \,d\Gamma$$ and similarly for the others?

What is the situation in 3D? In 3D the traction elements are bilinear quads (for trilinear hex elements). (Note: the faces of trilinear hexes in 3D may not be plane. See this.)Is the connectivity of traction elements in 3D defined so that the normal points outwards according to the right hand rule? So for the right most face element in the above figure, should the connectivity be (3,7,6,2) and for the topmost face element the connectivity should be (7,8,5,6)?

• I'm pretty sure that the direction of integration is immaterial. For 2D elasticity, I think that the boundary term $(w,h)_\Gamma_h$ is a line integral. The definition of a line integral renders the direction of integration immaterial. In 3D, the boundary term is a surface integral and I think it must be defined in a manner analogous to the line integral. – Nachiket Sep 8 '20 at 5:57