On finite element analysis of an inverse problem in elasticity

Abstract

This investigation concerns an inverse problem modelled by the finite element method. For a given mesh and set of physical properties, even though a well-posed direct problem possesses a unique solution in classical linear elasticity, variational arguments establish that a corresponding inverse problem may not. Furthermore, even when the inverse problem admits a unique solution when modelled ‘exactly’ using the classical linear theory of elasticity, an unfortunate mesh choice may cause the finite element model of the inverse problem to fail to do so. The current investigation introduces a simple matrix nonsingularity criterion assuring that the finite element model possesses a unique solution. An apparently new and readily computed numerical test is introduced to verify satisfaction of the criterion. Examples are given illustrating the effectiveness of the criterion and its application to mesh design.