Question

Upper and lower boundary conditions fem'

Answer 1

It is well known that the finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems. However, it is much more difficult to bound the solution from above for general problems in elasticity, and it has been a dream of many decades to find a systematical way to obtain such an upper bound. This paper presents a very important and unique property of the linearly conforming point interpolation method (LC-PIM): it provides a general means to obtain an upper bound solution in energy norm for elasticity problems. This paper conducts first a thorough theoretical study on the LC-PIM: we derive its weak form based on variational principles, study a number of properties of the LC-PIM, and prove that LC-PIM is variationally consistent and that it produces upper bound solutions. We then demonstrate these properties through numerical studies via examples of ID, 2D and 3D problems. Using the LC-PIM together with the FEM, we now have a systematically way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems.