On the accuracy of shape sensitivity

The precision with which the stress intensity factor (SIF) can be calculated from a finite element solution depends essentially on the extraction method and on the discretization error. In this paper, the influence of the discretization error in the SIF calculation was studied and a method for estimating the resulting error was developed. The SIF calculation method used is based on a shape design sensitivity analysis; this assures that the resulting error in the extracted SIF depends solely on the global discretization error present in the finite element solution. Moreover, this method allows us to extend the Zienkiewicz-Zhu discretization error estimator to the SIF calculation. The reliability of the proposed method was analysed solving a two-dimensional problem using an h-adaptive process. Also the convergence of the error with the h-adaptive refinement was studied. NOMENCLATURE a = crack length a, = design variable D = stress-strain relationship matrix (elasticity matrix) E, E' = elasticity modulus (plane stress and plane strain) 11 ecXfu, 11, /I eol(u) 11 = energy norm (and its estimation) of the exact error in displacements e,,(#,, ecx(K,), e,,,, e,,,,) = exact error in Y (in K,) and estimated error in Y (in K,) F = equivalent applied nodal force vector Y-, Yfc, Yes = strain energy release rate (exact, finite element and estimated solutions) h = element size IJI =determinant of the Jacobian matrix (isoparametric elements) K = stiffness matrix ne = number of elements p = degree of the complete polynomial used in the displacement interpolation N = number of degrees of freedom R = ratio of qa(, to q&") U =nodal displacement vector Kkx, Kwe, K,, = stress intensity factor, mode I, (exact, finite element and estimated solutions) hx, ufs = displacement field (exact and finite element solutions) II h x 11, II Ufe II =energy norm of 4 . x and of Ufe 11 hs 11 = estimated energy norm of u, q,,(,), vex(*, qes(u), qoa(g) =exact relative error in u (in 9) and estimated relative error in u (in 9) qM(Kl), qax(KIw, = estimated relative error in K , and exact relative error in K,, &, B, , , B, , , , = effectivity index of the error in u, in 9 and in K, I = constant which characterizes the intensity of a singularity v = Poisson ratio 5, = sensitivity of the squared energy norm with respect to a, l 7 = total potential energy tex, tfa = sensitivity of the squared energy norm (exact and finite element solutions) uex, ufc = stress field (exact solution and finite element solution) 813 814 J. FUENMAYOR et al. u* = improved stress field = convergence rate for qcx(yI) 62, 62, =domain on which the problem is defined and local domain of one finite element

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“…as a whole [ 191. Others, compare the sensitivity errors in terms of the mesh refinement effect [20,21].…”

Section: Shape Sensitivity Analysis As Applied To Sif Calculationmentioning

confidence: 99%

Calculation of the Stress Intensity Factor and Estimation of Its Error by a Shape Sensitivity Analysis

Fuenmayor

Domínguez

Giner

et al. 1997

Fatigue Fract Eng Mat Struct

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“…From this definition for the absolute error in sensitivities, the exact relative error can be defined as " e (12) In order to estimate the sensitivity discretization error, we thought it advisable to substitute equation (5) into (11). We thus had…”

Section: "#U#mentioning

confidence: 99%

Extension of the Zienkiewicz–zhu Error Estimator to Shape Sensitivity Analysis

Fuenmayor

Oliver

Ródenas

1997

Numerical Meth Engineering

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SUMMARYThe application of the Zienkiewicz-Zhu estimator was extended to the estimation of the discretization error arising from shape sensitivity analysis using the finite element method. The sensitivity error was quantified from the sensitivity of the energy norm by using an estimator specially developed for this purpose. Sensitivity analyses were carried out using the discrete analytical approach, which introduced no additional errors other than the discretization error. In this work, direct nodal averaging was used for linear triangular elements and the SPR technique for quadratic elements in order to obtain the smoothed stress and sensitivities fields. Two examples with an exact solution are used to analyse the effectivity of the proposed estimator and its convergence with the h-adaptive refinement.1997 by John Wiley & Sons, Ltd.

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“…However, in order to avoid difficulties arising when trying to exactly differentiate the second-order equations, some of the terms appearing in the gradient formulation are obtained in an approximate manner. This approach, also referred to as semianalytical, is already being used in structural optimization (Haftka and Barthelemy 1991) and has been the topic of recent work in aerodynamics (Svenningsten et al 1994). Emphasis is also put on the choice of objective functions and constraints representative of a real industrial problem, and the comprehensive use of mathematical programming tools allowing proper treatment of these constraints.…”

Section: Introductionmentioning

confidence: 98%

Constrained optimization of nacelle shapes in Euler flow using semianalytical sensitivity analysis

Lambert

Lecordix

Braibant

1995

Structural Optimization

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The problem of aerodynamic shape optimization in Euler flow is addressed. B-splines are used for parame:rization of the shape. Cheap gradient calculation is obtained via sensitivity analysis and the solution of an adjoint equation; pseudo secondorder spatial accuracy is achieved by means of a semianalytical folmulation. As a validation of the approach, several inverse and constrained optimization test problems are presented with emphasis on civil engine nacelle design. The handling of nondifferentiable quantities (such as maxima) in cost functions is allowed for via the use of the Kreisselmeier-Steinhauser function.

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On the accuracy of shape sensitivity

Cited by 17 publications

References 10 publications

Calculation of the Stress Intensity Factor and Estimation of Its Error by a Shape Sensitivity Analysis

Calculation of the Stress Intensity Factor and Estimation of Its Error by a Shape Sensitivity Analysis

Extension of the Zienkiewicz–zhu Error Estimator to Shape Sensitivity Analysis

Constrained optimization of nacelle shapes in Euler flow using semianalytical sensitivity analysis

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