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Jeff Borggaard - One of the best experts on this subject based on the ideXlab platform.
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The Sensitivity Equation method in fluid mechanics
European Journal of Computational Mechanics, 2008Co-Authors: Dominique Pelletier, Alexander Hay, Stephane Etienne, Jeff BorggaardAbstract:We present the Sensitivity Equation Method (SEM) as a complementary tool to adjoint based optimisation methods. Flow sensitivities exist independently of a design problem and can be used in several non-optimization ways: characterization of complex flows, fast evaluation of flows on nearby geometries, and input data uncertainties cascade through the CFD code to yield uncertainty estimates of the flow field. The Navier-Stokes and Sensitivity EquationsSensitivity are solved by an adaptive finite element method.
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A continuous Sensitivity Equation method for time-dependent incompressible laminar flows
International Journal for Numerical Methods in Fluids, 2006Co-Authors: Hristina G. Hristova, Stephane Etienne, Dominique Pelletier, Jeff BorggaardAbstract:We present a general Sensitivity Equation method (SEM) for time dependent incompressible laminar flows. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. The SEM formulation is verified on a problem with a closed form solution. Systematic grid convergence studies confirm the theoretical rates of convergence in both space and time. The methodology is then applied to pulsed flow around a square cylinder. The flow starts with symmetrical vortex shedding then transitions to the traditional Von Karman street (alternate vortex shedding). Simulations indicate that the transition phase manifests itself earlier in the Sensitivity fields than in the flow field itself. Sensitivities are then demonstrated for fast evaluation of nearby flows and uncertainty analysis.
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An improved continuous Sensitivity Equation method for optimal shape design in mixed convection
Numerical Heat Transfer Part B: Fundamentals, 2006Co-Authors: Régis Duvigneau, Dominique Pelletier, Jeff BorggaardAbstract:This article presents an optimal shape design methodology for mixed-convection problems. The Navier-Stokes Equations and the continuous Sensitivity Equations (CSEs) are solved using an adaptive finite-element method to obtain flow and Sensitivity fields. A new procedure is presented to extract accurate values of the flow derivatives at the boundary, appearing in the CSE boundary conditions for shape parameters. Flow and Sensitivity information are then employed to calculate the value and gradient of a design objective function. A Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm is used to find optimal shape parameter values. The proposed approach is first verified on a problem with a closed-form solution, obtained by the method of manufactured solutions. The method is then applied to determine the optimal shape of a model cooling system.
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A Second-order Sensitivity Equation method for laminar flow
International Journal of Computational Fluid Dynamics, 2005Co-Authors: J.-n. Mahieu, Stephane Etienne, Dominique Pelletier, Jeff BorggaardAbstract:This paper presents a general Continuous Sensitivity Equation (CSE) method for computing first and second-order flow sensitivities for the incompressible Navier–Stokes Equations. The Sensitivity Equations and boundary conditions are developed for value parameters that do not affect the geometry of the computational domain. Applications are thus restricted to the simpler case of value parameters. The flow and Sensitivity Equations are solved by an adaptive finite element method. The proposed methodology is verified on a problem with a closed form solution. The verified code is then applied to compute first- and second-order flow sensitivities of an airfoil with respect to the angle of attack and the free-stream velocity. We demonstrate the use of sensitivities for fast first- and second-order evaluations of nearby flows. The methodology is also used to perform second-order uncertainty analysis.
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a general continuous Sensitivity Equation formulation for the k e model of turbulence
International Journal of Computational Fluid Dynamics, 2004Co-Authors: E. Turgeon, Dominique Pelletier, Jeff BorggaardAbstract:In this paper, we develop a general formulation of the continuous Sensitivity Equations (CSEs) for the standard k- \epsilon model of turbulence with wall functions. The development is performed for value parameters that do not affect the geometry of the computational domain. The formulation accounts for complex parameter dependencies and results in the development of software that is suitable for a wide range of problems. In addition to details of an implementation within an existing adaptive finite element program, we perform a careful verification study and present an application of Sensitivity analysis to turbulent flow over a flat plate.
Dominique Pelletier - One of the best experts on this subject based on the ideXlab platform.
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A Continuous Sensitivity Equation of Arbitrary High Order
22nd AIAA Computational Fluid Dynamics Conference, 2015Co-Authors: Corinne Belley, Alexander Hay, Dominique PelletierAbstract:We present an approach to automatically generate and solve the flow sensitivities with respect to a given single parameter up to an arbitrary order n. We use the Newton multinomial theorem to automatically derive the set of terms constituting the Sensitivity Equations of any order. Hence, given the flow Equations at hand (Navier-Stokes, RANS, Burgers, etc), our methodology automatically produces the corresponding Equations for the flow sensitivities of an arbitrary high order n. In our approach, the flow and Sensitivity Equations are not calculated by different solvers resulting from different source codes. Rather, we extend an existing flow solver by adding an extra loop over the Sensitivity order (i.e. from 0 to n, the 0 order flow Sensitivity being the flow itself) on top of the main solution procedure. Thus, during the execution of the loop the first iteration computes the flow as before and the next iterations compute the flow sensitivities up to the requested order n. We present the necessary generic data structure to do so. The verification of the flow-and-Sensitivity solver is performed by the method of the manufactured solution. The computed sensitivities are validated by comparison to sensitivities obtained by second-order finite-differences. Finally, we examine the ability of high-order Taylor series expansions in parameter space to approximate flow solutions over a wide range of parameter values.
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A continuous Lagrangian Sensitivity Equation method for incompressible flow
Journal of Computational Physics, 2012Co-Authors: Lise Charlot, Stephane Etienne, Dominique PelletierAbstract:A continuous Lagrangian Sensitivity Equation method (CLSEM) is presented as a cost effective alternative to the continuous (Eulerian) Sensitivity Equation method (CESEM) in the case of shape parameters. Boundary conditions for the CLSEM are simpler than those of the CESEM. However a mapping must be introduced to relate the undeformed and deformed configurations thus making the PDEs more complicated. We propose the use of pseudo-elasticity Equations to provide a general framework to generate this mapping for unstructured meshes on complex geometries. The methodology is presented in details for the incompressible Navier-Stokes and Sensitivity Equations in variational form. The PDEs are solved with an adaptive FEM. Sensitivity data obtained with both approaches for a flow around a NACA 4512 are used to obtain estimates of flows around nearby geometries. Results indicate that the CLSEM produces significant improvements in terms of both accuracy and CPU time.
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first and second order Sensitivity Equation methods for value and shape parameters
International Journal for Numerical Methods in Fluids, 2008Co-Authors: Florin Ilinca, Dominique Pelletier, Alexander HayAbstract:This paper presents formulations of the Sensitivity Equation method (SEM) and applications to transient flow problems. Solutions are shown for both value and shape parameters using a three-dimensional solution algorithm. Sensitivities are used for fast evaluation of the flow at nearby values of the parameters: the solution is approximated by a Taylor series in parameter space involving the flow sensitivities. The accuracy of nearby flows is much improved when second-order sensitivities are used. We show how the Sensitivity of the Strouhal number can be obtained from the flow sensitivities. Results are in agreement with the experimental correlation. The methodology is also applied to the flow past a cylinder in ground proximity. The proposed method is verified on a steady-state problem by comparing the computed Sensitivity with the actual change in the solution when a small perturbation is imposed on the shape parameter. We then investigate the ability of the SEM to anticipate the unsteady flow response to changes in the ground to cylinder gap. The approach properly reproduces the damping or amplification of the vortex shedding with a reduction or increase of the gap size.
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The Sensitivity Equation method in fluid mechanics
European Journal of Computational Mechanics, 2008Co-Authors: Dominique Pelletier, Alexander Hay, Stephane Etienne, Jeff BorggaardAbstract:We present the Sensitivity Equation Method (SEM) as a complementary tool to adjoint based optimisation methods. Flow sensitivities exist independently of a design problem and can be used in several non-optimization ways: characterization of complex flows, fast evaluation of flows on nearby geometries, and input data uncertainties cascade through the CFD code to yield uncertainty estimates of the flow field. The Navier-Stokes and Sensitivity EquationsSensitivity are solved by an adaptive finite element method.
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A continuous Sensitivity Equation method for time-dependent incompressible laminar flows
International Journal for Numerical Methods in Fluids, 2006Co-Authors: Hristina G. Hristova, Stephane Etienne, Dominique Pelletier, Jeff BorggaardAbstract:We present a general Sensitivity Equation method (SEM) for time dependent incompressible laminar flows. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. The SEM formulation is verified on a problem with a closed form solution. Systematic grid convergence studies confirm the theoretical rates of convergence in both space and time. The methodology is then applied to pulsed flow around a square cylinder. The flow starts with symmetrical vortex shedding then transitions to the traditional Von Karman street (alternate vortex shedding). Simulations indicate that the transition phase manifests itself earlier in the Sensitivity fields than in the flow field itself. Sensitivities are then demonstrated for fast evaluation of nearby flows and uncertainty analysis.
Akbar Esfandiari - One of the best experts on this subject based on the ideXlab platform.
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model updating of a concrete beam with extensive distributed damage using experimental frequency response function
Journal of Bridge Engineering, 2016Co-Authors: Akbar Esfandiari, Alireza Rahai, Masoud Sanayei, Firooz BakhtiarinejadAbstract:An experimental verification of a parameter estimation algorithm for finite-element (FE) model updating for damage quantification of a concrete beam with extensive damage is presented using the measured frequency response function (FRF) data. The intact stage of the beam and four progressive damage stages are investigated to update the FE model to quantify the damage across the beam. The method uses a subset of measured FRF data for FE model updating via a quasi-linear Sensitivity Equation of structural response. A least-squares algorithm with an appropriate weighting technique is used for parameter estimation. Proper selection of excitation frequency ranges for model updating is addressed. Predicted equivalent stiffnesses demonstrate the success and robustness of the method in damage quantification, even when using only a subset of the measured FRF data.
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a frequency response based structural damage identification using model updating method
Structural Control & Health Monitoring, 2016Co-Authors: Fariba Shadan, Faramarz Khoshnoudian, Akbar EsfandiariAbstract:Summary Structural model updating by estimation of stiffness and mass parameters via monitoring of dynamic characteristics has attracted much attention in recent decades. In this study, frequency response functions (FRF) are utilized in order to identify unknown structural parameters using a Sensitivity-based model updating approach. A Sensitivity Equation that diminishes adverse effects of incompleteness of FRF data is proposed for model updating. Efficiency of the proposed method and impacts of measurement errors and incompleteness of measured data are examined numerically through a truss reference example. The stiffness and mass parameters of the intact model are updated using the damped FRFs of the simulated damaged model. The results demonstrate that the proposed method is capable of precisely identifying the location and the severity of damage in all studied cases. Copyright © 2015 John Wiley & Sons, Ltd.
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Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved Sensitivity Equation
International Journal of Mechanical Sciences, 2013Co-Authors: Akbar Esfandiari, Firooz Bakhtiari-nejad, Alireza RahaiAbstract:Abstract In this study, a frequency-based technique is presented to detect any number of localized damages which induce stiffness reduction in a structure. The Sensitivity of natural frequencies is characterized as a second order element level function of the stiffness reduction. To achieve accurate results, change in the mode shapes of the structure are expressed as a linear combination of the intact structure mode shapes and are considered to develop Sensitivity Equations. The overall formulation yields an indeterminate set of Equations. An optimization criterion is used to solve these Equations to obtain the change of the structural parameters. The method was applied to a truss using numerically simulated data and it was verified using experimentally measured data of a steel cantilever beam and a one-story one-bay frame. Results show the efficiency of the proposed method and the importance of including the changes of the mode shapes through a frequency-based damage detection algorithm and also approve that the method is able to detect damage without an exact model of the undamaged structure.
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structural model updating using frequency response function and quasi linear Sensitivity Equation
Journal of Sound and Vibration, 2009Co-Authors: Akbar Esfandiari, Firooz Bakhtiarinejad, Alireza Rahai, Masoud SanayeiAbstract:A method is presented for structural mass and stiffness estimation including damping effects and using vibration data. It uses the frequency response function (FRF) and natural frequencies data for finite element model updating. The FRF data are compiled using the measured displacement, velocity or acceleration of the damaged structure. A least-square algorithm method with appropriate normalization is used for solving the over-determined system of Equations with noise-polluted data. Sensitivity Equation normalization and proper selection of measured frequency points improved the accuracy and convergence in finite element model updating. Using simulated measurements shows that this method can detect, locate and quantify the severity of damage within structures.
John A. Burns - One of the best experts on this subject based on the ideXlab platform.
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A PDE Sensitivity Equation Method for Optimal Aerodynamic Design
Journal of Computational Physics, 1997Co-Authors: Jeff Borggaard, John A. BurnsAbstract:The use of gradient-based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design. A typical procedure uses a simulation scheme to evaluate the objective function (from the approximate states) and its gradient, then passes this information to an optimization algorithm. Once the simulation scheme (CFD flow solver) has been selected and used to provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients. Although this black-box approach has many advantages in shape optimization problems, one must compute mesh sensitivities in order to compute the design Sensitivity. In this paper, we present an alternative approach using the PDE Sensitivity Equation to develop algorithms for computing gradients. This approach has the advantage that mesh sensitivities need not be computed. Moreover, when it is possible to use the CFD scheme for both the forward problem and the Sensitivity Equation, then there are computational advantages. An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination of discretization schemes, one can showasymptotic consistencyunder mesh refinement, which is often sufficient to guarantee convergence of the optimal design algorithm. In particular, we show that when asymptotically consistent schemes are combined with a trust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as theSensitivity Equation method.The Sensitivity Equation method is presented, convergence results are given, and the approach is illustrated on two optimal design problems involving shocks.
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A Sensitivity Equation APPROACH TO SHAPE OPTIMIZATION IN FLUID FLOWS
Flow Control, 1995Co-Authors: Jeff Borggaard, John A. BurnsAbstract:In this paper we apply a Sensitivity Equation method to shape optimization problems. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential Equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.
Alireza Rahai - One of the best experts on this subject based on the ideXlab platform.
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model updating of a concrete beam with extensive distributed damage using experimental frequency response function
Journal of Bridge Engineering, 2016Co-Authors: Akbar Esfandiari, Alireza Rahai, Masoud Sanayei, Firooz BakhtiarinejadAbstract:An experimental verification of a parameter estimation algorithm for finite-element (FE) model updating for damage quantification of a concrete beam with extensive damage is presented using the measured frequency response function (FRF) data. The intact stage of the beam and four progressive damage stages are investigated to update the FE model to quantify the damage across the beam. The method uses a subset of measured FRF data for FE model updating via a quasi-linear Sensitivity Equation of structural response. A least-squares algorithm with an appropriate weighting technique is used for parameter estimation. Proper selection of excitation frequency ranges for model updating is addressed. Predicted equivalent stiffnesses demonstrate the success and robustness of the method in damage quantification, even when using only a subset of the measured FRF data.
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Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved Sensitivity Equation
International Journal of Mechanical Sciences, 2013Co-Authors: Akbar Esfandiari, Firooz Bakhtiari-nejad, Alireza RahaiAbstract:Abstract In this study, a frequency-based technique is presented to detect any number of localized damages which induce stiffness reduction in a structure. The Sensitivity of natural frequencies is characterized as a second order element level function of the stiffness reduction. To achieve accurate results, change in the mode shapes of the structure are expressed as a linear combination of the intact structure mode shapes and are considered to develop Sensitivity Equations. The overall formulation yields an indeterminate set of Equations. An optimization criterion is used to solve these Equations to obtain the change of the structural parameters. The method was applied to a truss using numerically simulated data and it was verified using experimentally measured data of a steel cantilever beam and a one-story one-bay frame. Results show the efficiency of the proposed method and the importance of including the changes of the mode shapes through a frequency-based damage detection algorithm and also approve that the method is able to detect damage without an exact model of the undamaged structure.
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structural model updating using frequency response function and quasi linear Sensitivity Equation
Journal of Sound and Vibration, 2009Co-Authors: Akbar Esfandiari, Firooz Bakhtiarinejad, Alireza Rahai, Masoud SanayeiAbstract:A method is presented for structural mass and stiffness estimation including damping effects and using vibration data. It uses the frequency response function (FRF) and natural frequencies data for finite element model updating. The FRF data are compiled using the measured displacement, velocity or acceleration of the damaged structure. A least-square algorithm method with appropriate normalization is used for solving the over-determined system of Equations with noise-polluted data. Sensitivity Equation normalization and proper selection of measured frequency points improved the accuracy and convergence in finite element model updating. Using simulated measurements shows that this method can detect, locate and quantify the severity of damage within structures.
