[joseph11554]

in the finite elements (CAE) there are a lot of way for achieving values between nodes but is it true that all equation for achieving values between nodes, indeed find the equation of a line or a plane that cross the nodes and if we give a position to that equation we could get the values?
so for triangular element if we could find a plane that cross the three nodes, we could achieve values between nodes.
if so what happened for 3D element?

[mprinkey]

Originally Posted by joseph11554

in the finite elements (CAE) there are a lot of way for achieving values between nodes but is it true that all equation for achieving values between nodes, indeed find the equation of a line or a plane that cross the nodes and if we give a position to that equation we could get the values?
so for triangular element if we could find a plane that cross the three nodes, we could achieve values between nodes.
if so what happened for 3D element?

I am not an FEA expert, but everything in FE (and Discontinuous Galkerin and Spectral Elements) is based on the shape functions (aka the basis functions). You may define compatibility conditions among elements however you like and you may choose the basis functions as you prefer, but once you have made those choices, whatever DOFs you use to define the basis completely determines the values of your functions within each element.

In 2D, you may have a triangular element with DOFs as function values at the three corners. You choose a simple linear basis in the element as phi(x,y) = A + B*x + C*y. If you have a line crossing that element and you want the variation of phi along that line you just LITERALLY substitute in--assume the line is y = m*x + b. Phi along that line is then phi(x) = A +B*x + C*(m*x + b). You can reparametrize that in terms of line length of whatever.

In 3D, it works exactly the same way. phi(x,y,z) = A + B*x + C*y + D*z on the tetrahedral element such that A,B,C,D all are consistent with the nodal values. Assume a plane s*x + t*y - z + u = 0 that passes through the tetrahedra. Phi on the plane cutting the tetrahedron is phi(x,y) = A + B*x + C*y + D*(s*x+t*y+u). Now, I've cheated a bit and made a plane that is easy to parametrize in only x and y, but the same approach works for any plane and any parametrization.