Numerical methods for second‐order shape sensitivity analysis with applications to heat conduction problems

Hou, Gene; Sheen, J. S.

doi:10.1002/nme.1620360305

Cited by 30 publications

(4 citation statements)

References 11 publications

(2 reference statements)

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“…Even in disciplines other than aerodynamics, the relevant literature is quite poor. The Hessian matrix computation using all possible discrete approaches for structural optimization has been presented in [123], while continuous approaches for the Hessian computation for heat conduction problems can be found in [124]. A different view of the same problem can be found in [125][126][127], for variational data assimilation problems in meteorology with the shallowwater equations as state equations.…”

Section: Direct Adjoint and Mixed Approaches In Aerodynamic Shape Opmentioning

confidence: 99%

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

Papadimitriou

Giannakoglou

2008

Arch Computat Methods Eng

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This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.

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Section: Direct Adjoint and Mixed Approaches In Aerodynamic Shape Opmentioning

confidence: 99%

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

Papadimitriou

Giannakoglou

2008

Arch Computat Methods Eng

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“…The concept of SSA plays a critical role as a gradient to guide parameter updates when performing iterative shape optimization to meet various physical performance requirements . Various work has shown how to build first‐order and second‐order shape sensitivities with respect to one or two shape control parameters . We use the concept of second‐order shape sensitivity to estimate defeaturing error when removing multiple features.…”

Section: Basics and Overviewmentioning

confidence: 99%

Second‐order defeaturing error estimation for multiple boundary features

Zhang

Martin

2014

Numerical Meth Engineering

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SUMMARYA posteriori estimation of defeaturing error, that is, engineering analysis error caused by defeaturing, remains a key challenge in computer-aided design/computer-aided engineering integration; indeed, it is essential if analysis based on automatic simplification of the geometry of a complex design is to be reliable. Previous work has mainly focused on removing a single, negative, internal design feature (i.e. a void within the model's interior); effects of removal of multiple features are found by simple summation. In this paper, we give a second-order defeaturing error estimator that can handle multiple boundary features, either positive or negative (cutouts or additions on the model's boundary). This is the first such method to take into account interactions between different features. (Removing internal holes in 2D or through holes in 3D will change the model's topology and is not addressed here.) The proposed estimator is also the first to handle positive features without heuristics. Our approach uses second-order shape sensitivity, on the basis of reformulating defeaturing as a shape transformation process. The reference model used for sensitivity computation is carefully chosen, as is the associated shape variation velocity, to ensure efficient computation and simple sensitivity expressions. Mixed partial derivatives in a second-order Taylor expansion are used to account for interaction between features. The advantages of our approach are demonstrated using various numerical examples.

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“…There are only a few works in the literature on the computation of second-order objective function sensitivities in optimization problems. The Hessian matrix computation for structural optimization has been presented in [35], whereas continuous approaches for the Hessian computation for heat conduction can be found in [36]. The computation of the condition number of the Hessian matrix is discussed in [32].…”

Section: Computation Of the Derivatives Of Fmentioning

confidence: 99%

“…In other than aerodynamic disciplines, a few more relevant works can be found. The Hessian matrix computation for structural optimization has been presented in [35], whereas continuous approaches for the Hessian computation for heat conduction can be found in [36]. In [37] and [38], the second-order sensitivities are computed for variational data assimilation problems in meteorology.…”

Section: Computation Of the Derivatives Of Fmentioning

confidence: 99%

Robust design in aerodynamics using third‐order sensitivity analysis based on discrete adjoint. Application to quasi‐1D flows

Papoutsis‐Kiachagias

Papadimitriou

Giannakoglou

2011

Numerical Methods in Fluids

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SUMMARY In this paper, the second‐order second moment approach, coupled with an adjoint‐based steepest descent algorithm, for the solution of the so‐called robust design problem in aerodynamics is proposed. Because the objective function for the robust design problem comprises first‐order and second‐order sensitivity derivatives with respect to the environmental parameters, the application of a gradient‐based method , which requires the sensitivities of this function with respect to the design variables, calls for the computation of third‐order mixed derivatives. To compute these derivatives with the minimum CPU cost, a combination of the direct differentiation and the discrete adjoint variable method is proposed. This is presented for the first time in the relevant literature and is the most efficient among other possible schemes on condition that the design variables are much more than the environmental ones; this is definitely true in most engineering design problems. The proposed approach was used for the robust design of a duct, assuming a quasi‐1D flow model; the coordinates of the Bézier control points parameterizing the duct shape are used as design variables, whereas the outlet Mach number and the Darcy–Weisbach friction coefficient are used as environmental ones. The extension to 2D and 3D flow problems, after developing the corresponding direct differentiation and adjoint variable methods and software, is straightforward. Copyright © 2011 John Wiley & Sons, Ltd.

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Numerical methods for second‐order shape sensitivity analysis with applications to heat conduction problems

Cited by 30 publications

References 11 publications

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

Second‐order defeaturing error estimation for multiple boundary features

Robust design in aerodynamics using third‐order sensitivity analysis based on discrete adjoint. Application to quasi‐1D flows

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