Bemtdy–the Boundary Element Analysis Program for 3-D Transient Dynamic Field

Zhou, Xu; Chen, H.K.

doi:10.1016/b978-0-08-040200-0.50026-1

Cited by 6 publications

(7 citation statements)

References 3 publications

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“…From (5), we have that ∂w n ∂n is bounded in L 2 (∂ ). Due to the shift theorem [8], the function w n is bounded independently of n in W 3 2 ,2 ( ). Using again lemma 2, the sequence {c n } is also bounded independently of n. Consequently, there exists a constant γ > 0 independent of n such that…”

Section: Theorem 1 the Optimization Problem Has A Global Minimum ψmentioning

confidence: 99%

Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem

2004

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We investigate an optimization problem (OP) in a non-standard form: the cost functional F measures the L 1 distance between the solution u ϕ of the direct Robin problem and a function f ∈ L 1 (M). After proving positivity, monotonicity and control properties of the state u ϕ with respect to ϕ, we prove the existence of an optimal control ψ to the problem (OP) and establish Newton differentiability of the functional F. As an application to this optimization problem the inverse problem of determining a Robin parameter ϕ inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of u ϕ inv . In spite of the fact that we work with the L 1 -norm we prove differentiability of the cost functional F by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to ϕ. An identifiability result is also proved for the set of admissible parameters ad consisting of positive functions in L ∞ .

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Section: Theorem 1 the Optimization Problem Has A Global Minimum ψmentioning

confidence: 99%

Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem

2004

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“…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 will be 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 [28,125] and the boundary element method [23,40]. The important role of the variational principles of elasticity, which in particular provide the foundations of the above-mentioned numerical solution methods, will be highlighted throughout this review.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems in elasticity

Bonnet¹,

2005

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Abstract. 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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“…We give a brief summary how to characterize the potential part of a field as a solution of an elliptic boundary value problem and how to solve this problem with the help of boundary element methods (BEM). For details we refer to [5] and Chen and Zhou [4].…”

Section: Potential Part As Solution Of An Elliptic Boundary Value Pro...mentioning

confidence: 99%

“…with Dirac's delta distribution δ(x), we can identify p | as the solution of an integral equation of second kind, see e.g. Chen and Zhou [4],…”

Section: Potential Part As Solution Of An Elliptic Boundary Value Pro...mentioning

confidence: 99%

Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method

Schuster¹

2004

Inverse Problems

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Vector field tomography deals with the reconstruction of velocity fields from line integrals over certain components of the field, the so-called Doppler transform. Even if we consider incompressible fluids, our reconstruction contains a potential part due to reconstruction errors and the null space of the Doppler transform which consists of potential fields. In this paper we present a method of defect correction for this sort of tomography detecting the potential part of the reconstruction by solving a boundary value problem. We solve this elliptic problem with the help of boundary element methods, where differentiation of the approximate solution can be avoided. If the reconstruction algorithm is based on the method of approximate inverse, the arising Newton potentials can be expressed using the Radon transform and are computed explicitly. The paper includes numerical results for solenoidal vector fields with tangential flow on the boundary from exact and noisy data.

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Bemtdy–the Boundary Element Analysis Program for 3-D Transient Dynamic Field

Cited by 6 publications

References 3 publications

Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem

Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem

Inverse problems in elasticity

Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method

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Bemtdy–the Boundary Element Analysis Program for 3-D Transient Dynamic Field

Cited by 6 publications

References 3 publications

Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem

Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem

Inverse problems in elasticity

Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method

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Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem

Differentiability properties of the L ¹ -tracking functional and application to the Robin inverse problem