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Tied Interfaces

Tied interfaces are composed of primary and constrained surfaces. Internally these are broken down to nodes on the constrained side and element (faces) on the primary side. The nodes on the constrained side are connected to the adjacent primary face via a set of constraint equations.

The (r,s)(r,s) coordinates of the nodes relative to the primary face are established and then the shape functions are used to construct a set of constraint equations

us(r,s)=∑ihiup,iu_{s}(r,s) = \sum_{i}^{}{h_{i}u_{p,i}}

In the case of a quad-4 face this expands to

us(r,s)=14(1−r)(1−s)up,1+14(1+r)(1−s)up,2+14(1+r)(1+s)up,3+14(1−r)(1+s)up,4u_{s}(r,s) = \frac{1}{4}(1 - r)(1 - s)u_{p,1} + \frac{1}{4}(1 + r)(1 - s)u_{p,2} + \frac{1}{4}(1 + r)(1 + s)u_{p,3} + \frac{1}{4}(1 - r)(1 + s)u_{p,4}

which forms the constraint equation. This is repeated for all the displacement directions.

The special case is the drilling degree of freedom. As the 2D elements have either no drilling freedom or one which can work quite locally. For this degree of freedom the rotation is linked to the translations of the 2D element. If the node is internal to the element base the rotation of the element as a whole. If the node is on the edge use the rotation of just that edge.

For each element node define a vector vi\mathbf{v}_{i} from the constrained node position to the node in the plane of the element. Let

li=∣vi∣l_{i} = \left| \mathbf{v}_{i} \right|
tan⁡αi=vi,yvi,x\tan\alpha_{i} = \frac{v_{i,y}}{v_{i,x}}

The displacement at the centre of the 2D element is

uc=∑ihiuiu_{c} = \sum_{i}^{}{h_{i}u_{i}}

Then the rotation of the node at a distance lil_{i} is and angle αi\alpha_{i}

θi=−(ui,x−uc,x)sin⁡αi+(ui,y−uc,y)cos⁡αili\theta_{i} = \frac{- \left( u_{i,x} - u_{c,x} \right)\sin\alpha_{i} + \left( u_{i,y} - u_{c,y} \right)\cos\alpha_{i}}{l_{i}}

So rotation at (r,s)(r,s) is

θ(r,s)=∑ihi−(ui,x−uc,x)sin⁡αi+(ui,y−uc,y)cos⁡αili\theta(r,s) = \sum_{i}^{}{h_{i}\frac{- \left( u_{i,x} - u_{c,x} \right)\sin\alpha_{i} + \left( u_{i,y} - u_{c,y} \right)\cos\alpha_{i}}{l_{i}}}

Or expanding

θ(r,s)=∑ihi−ui,xsin⁡αi+ui,ycos⁡αili−∑ihi−uc,xsin⁡αi+uc,ycos⁡αiliθ(r,s)=∑ihi−ui,xsin⁡αi+ui,ycos⁡αili−∑ihi−∑j(hjuj,x)sin⁡αi+∑j(hjuj,y)cos⁡αili\begin{aligned}\theta(r,s) &= \sum_{i}^{}{h_{i}\frac{- u_{i,x}\sin\alpha_{i} + u_{i,y}\cos\alpha_{i}}{l_{i}}} - \sum_{i}^{}{h_{i}\frac{- u_{c,x}\sin\alpha_{i} + u_{c,y}\cos\alpha_{i}}{l_{i}}}\\ \theta(r,s) &= \sum_{i}^{}{h_{i}\frac{- u_{i,x}\sin\alpha_{i} + u_{i,y}\cos\alpha_{i}}{l_{i}}} - \sum_{i}^{}{h_{i}\frac{- \sum_{j}^{}\left( h_{j}u_{j,x} \right)\sin\alpha_{i} + \sum_{j}^{}\left( h_{j}u_{j,y} \right)\cos\alpha_{i}}{l_{i}}}\end{aligned}