An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

2020 ◽  
Vol 13 (3) ◽  
pp. 689-718
Author(s):  
global sci
Keyword(s):  
Objective Function ◽  
Adjoint Method ◽  
Boundary Integral ◽  
Variational Technique ◽  
External Aerodynamics

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.

Three-dimensional optimum design of the cooling lines of injection moulds based on boundary element design sensitivity analysis

2002 ◽  
Vol 216 (7) ◽  
pp. 1067-1071 ◽  
Author(s):  
H Zhou ◽  
D Li ◽  
S Cui
Keyword(s):  
Sensitivity Analysis ◽  
Objective Function ◽  
Boundary Element ◽  
Optimum Design ◽  
Three Dimensional ◽  
Design Sensitivity ◽  
Boundary Integral ◽  
Optimization Techniques ◽  
Design Sensitivity Analysis ◽  
Design Variables

A three-dimensional numerical simulation using the boundary element method is proposed, which can predict the cavity temperature distributions in the cooling stage of injection moulding. Then, choosing the radii and positions of cooling lines as design variables, the boundary integral sensitivity formulations are deduced. For the optimum design of cooling lines, the squared difference between the objective temperature and the temperature of the cavity is taken as the objective function. Based on the optimization techniques with design sensitivity analysis, an iterative algorithm to reach the minimum value of the objective function is introduced, which leads to the optimum design of cooling lines at the same time.

Multi-Stage Turbomachinery Blades Optimization Design Using Adjoint Method and Thin Shear-Layer N-S Equations

2012 ◽  
Author(s):  
Lucheng Ji ◽  
Weiwei Li ◽  
Yong Tian ◽  
Weilin Yi ◽  
Jiang Chen
Keyword(s):  
Objective Function ◽  
Shear Layer ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Shape Design ◽  
Compressor Stage ◽  
Flow Equations ◽  
Aerodynamic Shape ◽  
Multi Stage ◽  
Turbomachinery Blades

Traditionally, 3D aerodynamic shape design with the aid of optimization algorithm in an analysis mode has provided a rational and direct search through design space, but it is usually too time-consuming. Further improvement to reduce design cycle is probably a necessary concern in turbomachinery community. Due to less computational cost, adjoint method has received considerable attention in recent years. This paper focuses on continuous adjoint method, and couples with thin shear-layer N-S equations to formulate an efficient sensitivity analysis model for multi-stage turbomachinery blades in the specified objective function. This model includes adjoint equations/boundary conditions, and the sensitivity of objective function to design variable vector. Integrating a 3D blade perturbation parameterization and the simple steepest decent method, a frame of a gradient-based aerodynamic shape design system is constructed. Numerical implementation to solve flow equations and adjoint equations is very similar, and once they are converged respectively, the sensitivity can be calculated by complex method and mesh perturbation efficiently. Thus, a fast Automatic-CFD-Design tool is developed, including three sub-solvers to solve flow equations, adjoint equations and calculate sensitivity respectively. Flow surface design of a 1-1/2 compressor stage in the specified target pressure distribution is used to validate the present approach. Flow field design of NASA transonic compressor stage 35 aiming to increase efficiency and remain mass flow rate and pressure ratio unchanged is taken.

A Variational Technique for Element Free Analysis of Static and Dynamic Fracture Mechanics

2010 ◽  
Vol 454 ◽  
pp. 31-46
Author(s):  
P.H. Wen ◽  
M.H. Aliabadi
Keyword(s):  
Stress Intensity ◽  
Fracture Mechanics ◽  
Stress Intensity Factors ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Inversion Method ◽  
Variational Technique ◽  
Intensity Factors ◽  
Element Free ◽  
Benchmark Solutions

. In this paper a variational technique is developed to calculate stress intensity factors with high accuracy using the element free Glerkin method. The stiffness and mass matrices are evaluated by regular domain integrals and the shape functions to determine displacements in the domain are calculated with radial basis function interpolation. Stress intensity factors were obtained by a boundary integral with a variation of crack length along the crack front. Based on a static reference solution, the transformed stress intensity factors in the Laplace space are obtained and Durbin inversion method is utilised in order to determine the physical values in time domain. The applications of proposed technique to two and three dimensional fracture mechanics are presented. Comparisons are made with benchmark solutions and indirect boundary element method.

scholarly journals Parametric Level Set-Based Multimaterial Topology Optimization of Heat Conduction Structures

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yadong Shen ◽  
Jianhu Feng
Keyword(s):  
Topology Optimization ◽  
Heat Conduction ◽  
Objective Function ◽  
Level Set ◽  
Heat Dissipation ◽  
Adjoint Method ◽  
Basis Functions ◽  
Expansion Coefficients ◽  
Quadratic Temperature ◽  
Moving Asymptotes

This paper presents a parametric level set-based method (PLSM) for multimaterial topology optimization of heat conduction structures with volume constraints. A parametric level set-based optimization model of heat conduction structures is built with multimaterial level set (MM-LS) model, which describes the boundaries of different materials by the combination of all level set functions. The heat dissipation efficiency which means the quadratic temperature gradient is conducted as the objective function. The adjoint method is utilized to calculate the sensitivities of the objective function with respect to expansion coefficients of the compactly supported radial basis functions (CSRBFs). The optimal configuration is achieved by updating the expansion coefficients gradually with the method of moving asymptotes (MMA). Several numerical examples are discussed to demonstrate effectiveness of the proposed method for multimaterial topology optimization of heat conduction structures.

Noise reduction in car aerodynamics using a surrogate objective function and the continuous adjoint method with wall functions

2015 ◽  
Vol 122 ◽  
pp. 223-232 ◽  
Author(s):  
E.M. Papoutsis-Kiachagias ◽  
N. Magoulas ◽  
J. Mueller ◽  
C. Othmer ◽  
K.C. Giannakoglou
Keyword(s):  
Objective Function ◽  
Noise Reduction ◽  
Adjoint Method ◽  
Wall Functions ◽  
Continuous Adjoint Method ◽  
Continuous Adjoint

scholarly journals Aerodynamic Shape Optimization of a Pipe Using the Adjoint Method

2012 ◽  
Author(s):  
Eysteinn Helgason ◽  
Siniša Krajnović
Keyword(s):  
Pressure Drop ◽  
Flow Field ◽  
Shape Optimization ◽  
Objective Function ◽  
Turbulence Model ◽  
Total Pressure ◽  
Adjoint Method ◽  
Inlet Pipe ◽  
Wall Functions ◽  
Flow Solver

Shape optimization of an inlet pipe to an engine re-circulator cooler using the adjoint method is presented. The method uses surface sensitivities calculated from an adjoint flow field implemented in the finite volume CFD solver OpenFOAM® [1]. This method allows for computation of the whole sensitivity field with only two solver calls, a primal and an adjoint solver call. A RANS solver with the standard k-epsilon turbulence model applying standard wall functions was used for the primal flow solver. The adjoint surface sensitivities are calculated from the adjoint and the primal flow fields and give information about how the objective function is affected by normal motion of the surface. The surface sensitivities are coupled to a mesh morphing library in OpenFOAM diffusing the motion of the boundary nodes to the internal cells of the mesh. The resulting geometry gave a 6.5% decrease in the total pressure drop through the pipe.

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.

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