Inverse Problem for Isentropic Mach-Number on Blade Wall in Aerodynamic Shape Design of Turbomachinery Cascades by Using Adjoint Method

2011 ◽  
Author(s):  
Haitao Li ◽  
Liming Song ◽  
Pengfei Zhang ◽  
Zhenping Feng
Keyword(s):  
Inverse Problem ◽  
Mach Number ◽  
Objective Function ◽  
Adjoint System ◽  
Boundary Integral ◽  
Adjoint Equations ◽  
Grid Node ◽  
Time Step ◽  
Flow Quantity ◽  
Continuous Adjoint

This paper presents an approach of the continuous adjoint system deduction based on the variation in grid node coordinates, in which the variation in the gradient of flow quantity is converted into the gradient of the variation in flow quantity and the gradient of the variation in grid node coordinates, which avoids the coordinate system transforming in the traditional derivation process of adjoint system and make the adjoint system much more sententious. By introducing the Jacobian matrix of viscous flux to the gradient of flow variables, the adjoint system for turbomachinery aerodynamic design optimization governed by compressible Navier-Stokes equations is derived in details. Given the general expression of objective functions consisted of both boundary integral and field integral, the adjoint equations and their boundary conditions are derived, and the final expression of the objective function gradient including only boundary integrals is formulated to reduce the CPU cost. Then the adjoint system is numerically solved by using the finite volume method with an explicit 5-step Runge-Kutta scheme and Riemann approximate solution of Roe’s scheme combined with multi-grid technique and local time step to accelerate the convergence procedure. Finally, the application of the method is illustrated through a turbine cascade inverse problem with an objective function of isentropic Mach number distribution on the blade wall.

scholarly journals Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

2021 ◽  
Vol 2090 (1) ◽  
pp. 012139
Author(s):  
OA Shishkina ◽  
I M Indrupskiy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Objective Function ◽  
Petroleum Reservoirs ◽  
Forward Problem ◽  
Adjoint Equations ◽  
Well Testing ◽  
Adjoint Methods ◽  
Two Phase ◽  
Gradient Calculation

Abstract Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

2021 ◽  
Author(s):  
Fellcitas Schäfer ◽  
Luca Magri ◽  
Wolfgang Polifke
Keyword(s):  
Network Model ◽  
Hybrid Approach ◽  
Adjoint System ◽  
Scattering Matrix ◽  
Adjoint Equations ◽  
Jump Conditions ◽  
Discrete Adjoint ◽  
Direct Scattering ◽  
Annular Combustor ◽  
Continuous Adjoint

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

S2 Surface Quasi-3D Aerodynamic Design Using the Continuous Adjoint Method for Multi-Stage Turbine

2014 ◽  
Author(s):  
Lei Chen ◽  
Jiang Chen
Keyword(s):  
Objective Function ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Design System ◽  
Shape Design ◽  
Aerodynamic Design ◽  
Aerodynamic Shape ◽  
Multi Stage ◽  
Gradient Based ◽  
Continuous Adjoint

The adjoint method eliminates the dependence of the gradient of the objective function with respect to design variables on the flow field making the obtainment of the gradient both accurate and fast. For this reason, the adjoint method has become the focus of attention in recent years. This paper develops a continuous adjoint formulation for through-flow aerodynamic shape design in a multi-stage gas turbine environment based on a S2 surface quasi-3D problem governed by the Euler equations with source terms. Given the general expression of the objective function calculated via a boundary integral, the adjoint equations and their boundary conditions are derived in detail by introducing adjoint variable vectors. As a result, the final expression of the objective function gradient only includes the terms pertinent to those physical shape variations that are calculated by metric variations. The adjoint system is solved numerically by a finite-difference method with explicit Euler time-marching scheme and a Jameson spatial scheme which employs first and third order dissipative flux. Integrating the blade stagger angles and passage perturbation parameterization with the simple steepest decent method, a gradient-based aerodynamic shape design system is constructed. Finally, the application of the adjoint method is validated through a 5-stage turbine blade and passage optimization with an objective function of entropy generation. The result demonstrates that the gradient-based system can be used for turbine aerodynamic design.

scholarly journals Derivation of the Adjoint Drift Flux Equations for Multiphase Flow

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 31 ◽  
Author(s):  
Shenan Grossberg ◽  
Daniel S. Jarman ◽  
Gavin R. Tabor
Keyword(s):  
Multiphase Flow ◽  
Objective Function ◽  
Settling Velocity ◽  
Flow Problem ◽  
Adjoint System ◽  
Drift Flux ◽  
Adjoint Approach ◽  
Continuous Adjoint ◽  
First Time ◽  
Mathematical Derivation

The continuous adjoint approach is a technique for calculating the sensitivity of a flow to changes in input parameters, most commonly changes of geometry. Here we present for the first time the mathematical derivation of the adjoint system for multiphase flow modeled by the commonly used drift flux equations, together with the adjoint boundary conditions necessary to solve a generic multiphase flow problem. The objective function is defined for such a system, and specific examples derived for commonly used settling velocity formulations such as the Takacs and Dahl models. We also discuss the use of these equations for a complete optimisation process.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

2021 ◽  
Author(s):  
Felicitas Schaefer ◽  
Luca Magri ◽  
Wolfgang Polifke
Keyword(s):  
Network Model ◽  
Hybrid Approach ◽  
Adjoint System ◽  
Scattering Matrix ◽  
Adjoint Equations ◽  
Jump Conditions ◽  
Discrete Adjoint ◽  
Direct Scattering ◽  
Annular Combustor ◽  
Continuous Adjoint

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Adjoint-Based Optimization Method With Linearized SST Turbulence Model and a Frozen Gamma-Theta Transition Model Approach for Turbomachinery Design

2015 ◽  
Author(s):  
Pengfei Zhang ◽  
Juan Lu ◽  
Zhiduo Wang ◽  
Liming Song ◽  
Zhenping Feng
Keyword(s):  
Turbulence Model ◽  
Entropy Generation ◽  
Adjoint System ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Transition Model ◽  
Transition Flow ◽  
Flow Optimization ◽  
Sst Turbulence Model ◽  
Continuous Adjoint

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 loss 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.

scholarly journals Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 129
Author(s):  
Yuan Li ◽  
Ni Zhang ◽  
Yuejiao Gong ◽  
Wentao Mao ◽  
Shiguang Zhang
Keyword(s):  
Side Effect ◽  
Domain Boundary ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Integration Scheme ◽  
Computational Accuracy ◽  
Time Step ◽  
Corner Points ◽  
Discontinuous Elements ◽  
The Time Domain

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

scholarly journals An Advance-Tracing Boundary Element Method in the Time Domain for Solving the Radiating Problem of an Arbitrary Obstacle Accelerating Randomly

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Jui-Hsiang Kao
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Time Domain ◽  
Boundary Integral ◽  
Normal Velocity ◽  
Time Step ◽  
Random Acceleration ◽  
The Time Domain ◽  
First Time ◽  
Element Method

This research develops an Advance-Tracing Boundary Element Method in the time domain to calculate the waves that radiate from an immersed obstacle moving with random acceleration. The moving velocity of the immersed obstacle is multifrequency and is projected along the normal direction of every element on the obstacle. The projected normal velocity of every element is presented by the Fourier series and includes the advance-tracing time, which is equal to a quarter period of the moving velocity. The moving velocity is treated as a known boundary condition. The computing scheme is based on the boundary integral equation in the time domain, and the approach process is carried forward in a loop from the first time step to the last. At each time step, the radiated pressure on each element is updated until obtaining a convergent result. The Advance-Tracing Boundary Element Method is suitable for calculating the radiating problem from an arbitrary obstacle moving with random acceleration in the time domain and can be widely applied to the shape design of an immersed obstacle in order to attain security and confidentiality.

Inversion of induced polarization data

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1327-1341 ◽  
Author(s):  
Douglas W. Oldenburg ◽  
Yaoguo Li
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
Effective Conductivity ◽  
Optimization Theory ◽  
Induced Polarization ◽  
Dc Resistivity ◽  
Earth Structure ◽  
Nonlinear Inverse Problem ◽  
Polarization Data ◽  
The Earth

We develop three methods to invert induced polarization (IP) data. The foundation for our algorithms is an assumption that the ultimate effect of chargeability is to alter the effective conductivity when current is applied. This assumption, which was first put forth by Siegel and has been routinely adopted in the literature, permits the IP responses to be numerically modeled by carrying out two forward modelings using a DC resistivity algorithm. The intimate connection between DC and IP data means that inversion of IP data is a two‐step process. First, the DC potentials are inverted to recover a background conductivity. The distribution of chargeability can then be found by using any one of the three following techniques: (1) linearizing the IP data equation and solving a linear inverse problem, (2) manipulating the conductivities obtained after performing two DC resistivity inversions, and (3) solving a nonlinear inverse problem. Our procedure for performing the inversion is to divide the earth into rectangular prisms and to assume that the conductivity σ and chargeability η are constant in each cell. To emulate complicated earth structure we allow many cells, usually far more than there are data. The inverse problem, which has many solutions, is then solved as a problem in optimization theory. A model objective function is designed, and a “model” (either the distribution of σ or η)is sought that minimizes the objective function subject to adequately fitting the data. Generalized subspace methodologies are used to solve both inverse problems, and positivity constraints are included. The IP inversion procedures we design are generic and can be applied to 1-D, 2-D, or 3-D earth models and with any configuration of current and potential electrodes. We illustrate our methods by inverting synthetic DC/IP data taken over a 2-D earth structure and by inverting dipole‐dipole data taken in Quebec.

An Inverse Algorithm to Estimate Spatially Varying Thermal Contact Resistance

2002 ◽  
Author(s):  
Eduardo Divo ◽  
Alain J. Kassab ◽  
Jennifer Gill
Keyword(s):  
Inverse Problem ◽  
Contact Resistance ◽  
Objective Function ◽  
Thermal Contact Resistance ◽  
Thermal Contact ◽  
Random Noise ◽  
Surface Characteristics ◽  
Interfacial Region ◽  
Field Problem ◽  
Additional Information

Characterization of the thermal contact resistance is important in modeling of multi-component thermal systems which feature mechanically mated surfaces. Thermal resistance is phenomenologically quite complex and depends on many parameters including surface characteristics of the interfacial region and contact pressure. In general, the contact resistance varies as a function of pressure and is non-uniform along the interface. An inverse problem is formulated to estimate the variation of the contact resistance. A two-dimensional model is considered where the contact resistance is sought along the contact line at the interface between two regions. Temperature measured at discrete locations using embedded sensors placed in proximity to the interface provides the additional information required to solve the inverse problem. Given current estimates of the contact resistance as a function of position along the interface, a forward problem is solved, and a quadratic objective function is formulated to evaluate the difference between predicted temperatures at the sensors and those measured. A genetic algorithm is used to minimize the objective function and obtain the best estimate of the contact resistance. A boundary element method is used to solve the forward temperature field problem. Numerical simulations are carried out to demonstrate the approach. Random noise is used to simulate the effect of input uncertainties in measured temperatures at the sensors.

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