Upper and Lower Bounds for Numerical Solutions of Elasticity Problems using LC-PIM and FEM

G. R. Liu () and G. Y. Zhang ()
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G. R. Liu: National University of Singapore, Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering
G. Y. Zhang: National University of Singapore, Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering

A chapter in Computational Mechanics, 2007, pp 148-155 from Springer

Abstract: Abstract 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.

Keywords: linearly conforming; point interpolation method; upper and lower bounds in energy norm (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-75999-7_13

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DOI: 10.1007/978-3-540-75999-7_13

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