Adjoint-weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data (original) (raw)
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Many inverse problems are formulated and solved as optimization problems. In this approach, the data mismatch between a predicted field and a measured field is minimized, subject to a constraint. The constraint represents the "forward" model of the system under consideration. In this paper, the model considered is plane stress incompressible elasticity. This pde is discretized using several standard Galerkin finite element methods. These are known to yield stable and convergent discrete solutions that converge with mesh refinement to the exact solution of the forward problem. It is usually taken for granted that if the constraint equation is discretized by a stable, convergent numerical method, then the inverse problem will also converge to the exact solution with mesh refinement. We show examples in this paper, however, where this is not the case. These are based on inverse problems with interior data, which have provably unique solutions. Even so, the use of classical discretization techniques for the forward constraint within the optimization formulation leads to ill-posed discrete problems. We analyze the discrete systems of equations and show the source of the instability. We discuss variational properties of the continuous inverse optimization problem, and describe a novel B-spline FEM to solve it. We present computational evidence that suggests the B-spline FEM inverse problem solution converges to the exact inverse problem solution with mesh refinement.
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This article is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters, or buried objects such as cracks. These inverse problems are considered mainly for threedimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e. fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.
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