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Piotr Fedelinski – One of the best experts on this subject based on the ideXlab platform.

William W. Hager – One of the best experts on this subject based on the ideXlab platform.

  • Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions
    Journal of Optimization Theory and Applications, 2021
    Co-Authors: Joseph D. Eide, William W. Hager, Anil V. Rao
    Abstract:

    A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed Adjoint System of the modified Legendre–Gauss–Radau method is then developed. Using this transformed Adjoint System, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

  • Modified Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions Part II: Costate Estimation and the Transformed Adjoint System
    2018 IEEE Conference on Decision and Control (CDC), 2018
    Co-Authors: Joseph D. Eide, William W. Hager
    Abstract:

    A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed Adjoint System for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.

  • CDC – Modified Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions Part II: Costate Estimation and the Transformed Adjoint System
    2018 IEEE Conference on Decision and Control (CDC), 2018
    Co-Authors: Joseph D. Eide, William W. Hager, Anil V. Rao
    Abstract:

    A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed Adjoint System for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.

Tadeusz Burczyński – One of the best experts on this subject based on the ideXlab platform.

  • Boundary elements in shape design sensitivity analysis and optimal design of vibrating structures
    Engineering Analysis With Boundary Elements, 1991
    Co-Authors: Tadeusz Burczyński, Piotr Fedelinski
    Abstract:

    Abstract A general approach to shape design sensitivity analysis and optimal design for dynamic transient and free vibrations problems using boundary elements is presented. The material derivatives and the Adjoint System method are applied to obtain first-order sensitivities for the effect of boundary shape variations. A numerical example of shape sensitivity analysis and optimal design for free vibrations of an elastic body is presented.

Zhenping Feng – One of the best experts on this subject based on the ideXlab platform.

  • Study on continuous Adjoint optimization with turbulence models for aerodynamic performance and heat transfer in turbomachinery cascades
    International Journal of Heat and Mass Transfer, 2017
    Co-Authors: Pengfei Zhang, Liming Song, Zhenping Feng
    Abstract:

    Abstract A continuous Adjoint method for turbomachinery is presented based on the varied turbulence eddy viscosity (VEV), rather than the constant eddy viscosity (CEV) assumption. Firstly, the grid node coordinates variation and Jacobian Matrices is introduced to deduce the general Adjoint System. Then, an objective of entropy generation for aerodynamic and heat transfer is proposed to evaluate the loss of both flow and heat transfer. The VEV Adjoint Systems with Spalart-Allmaras and SST turbulence models are established for the compressible turbulent flow in turbine cascades with the adiabatic blade wall condition. The aerodynamic optimization cases for turbine cascades show that the VEV Adjoint System can achieve higher accuracy, quicker convergence and better optimal result than that of the CEV System in turbomachinery. Furthermore, the improvement of the VEV Adjoint method to the mass flow rate constraint is analyzed. Finally, the VEV Adjoint System with linearized turbulence model is presented for the isothermal blade wall condition. The optimization results demonstrate the ability of these Systems in optimizing the flow and heat transfer performance and reducing the turbine total loss.

  • Adjoint-Based Optimization Method With Linearized SST Turbulence Model and a Frozen Gamma-Theta Transition Model Approach for Turbomachinery Design
    Volume 2B: Turbomachinery, 2015
    Co-Authors: Pengfei Zhang, Liming Song, Zhiduo Wang, Zhenping Feng
    Abstract:

    In this paper, based on the grid node coordinates variation and Jacobian Matrices, the turbulent continuous Adjoint method with linearized turbulence model is studied and developed to fully account for the variation of turbulent eddy viscosity. The corresponding Adjoint equations, boundary conditions and the final sensitivities are formulated with a general expression. To implement the Adjoint optimization of the transition flow, a flow solver combining the transition model with the turbulence model is employed, and an Adjoint optimization framework with linearized SST turbulence model and a frozen Gamma-Theta transition model is established. In order to choose an appropriate objective for the transition flow optimization, four objectives are studied, including the entropy generation, the total pressure lossloss coefficient, the field integral of turbulent kinetic energy, the area ratio of transition and turbulent regions to the suciton side. And finally the entropy generation is adopted as the objective. Then, the derivation of the Adjoint System for the entropy generation optimization is presented. To demonstrate the validity of the Adjoint System for transition flow, four shape optimizations for the bypass transitions and the separation-induced transition are implemented. A 2D isentropic case for bypass transitions is conducted to compares the performances of the fully turbulent Adjoint System and the frozen Gamma-Theta transition Adjoint System, while the other isothermal case is performed to take the aerodynamic and heat transfer issues into account together. The case of separation-induced transition is performed and also consistent well with its flow mechanism. The four optimization results illustrate the effectiveness of the Adjoint System for the transition flow optimization, which can improves the performance of overall cascades and the transition region.Copyright © 2015 by ASME

  • Study and Verification of Continuous Adjoint System for Aerodynamic Optimization in Turbomachinery Cascades
    Volume 6B: Turbomachinery, 2013
    Co-Authors: Pengfei Zhang, Zhenping Feng
    Abstract:

    This paper is a further study of the authors’ previous work on the continuous Adjoint method based on the variation in grid node coordinates and Jacobi Matrices of the flow fluxes. This method simplifies the derivation and expression of the Adjoint System, and reduces the computation cost. In this paper, the differences between the presented and the traditional methods are analyzed in details by comparing the derivation processes and the Adjoint Systems. In order to demonstrate the reliability and accuracy of the Adjoint System deduced by the authors, the presented method is applied to different optimal problems, which include two inverse designs and two shape optimizations in both 2D and 3D cascades. The inverse designs are implemented by giving the isentropic Mach number distributions along the blade wall for 2D inviscid flow and 3D laminar flow. The shape optimizations are implemented with the objective function of the entropy generation in flow passage for 2D and 3D laminar flows. In the 3D optimal case, this method is validated by supersonic turbine design case with and without mass flow rate constraint. The numerical results testify the accuracy of this Adjoint method, which includes only the boundary integrals, and furthermore, the universality and portability of this Adjoint System for inverse designs and shape optimizations are demonstrated.Copyright © 2013 by ASME

Joseph D. Eide – One of the best experts on this subject based on the ideXlab platform.

  • Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions
    Journal of Optimization Theory and Applications, 2021
    Co-Authors: Joseph D. Eide, William W. Hager, Anil V. Rao
    Abstract:

    A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed Adjoint System of the modified Legendre–Gauss–Radau method is then developed. Using this transformed Adjoint System, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

  • Modified Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions Part II: Costate Estimation and the Transformed Adjoint System
    2018 IEEE Conference on Decision and Control (CDC), 2018
    Co-Authors: Joseph D. Eide, William W. Hager
    Abstract:

    A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed Adjoint System for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.

  • CDC – Modified Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions Part II: Costate Estimation and the Transformed Adjoint System
    2018 IEEE Conference on Decision and Control (CDC), 2018
    Co-Authors: Joseph D. Eide, William W. Hager, Anil V. Rao
    Abstract:

    A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed Adjoint System for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.