An implementation of adjoint-based topology optimization in magnetostatics

2019 ◽  
Vol 38 (3) ◽  
pp. 1023-1035 ◽  
Author(s):  
Rtimi Youness ◽  
Frederic Messine
Keyword(s):  
Topology Optimization ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Design Problem ◽  
Large Scale ◽  
Adjoint Method ◽  
Processing Unit ◽  
Original Design ◽  
Content Type

Purpose In magnetostatics, topology optimization (TO) addresses the problem of finding the distributions of both current densities and ferromagnetic materials to comply with fixed magnetic specifications. The purpose of this paper is to develop TO in order to design Hall-effect Thrusters (HETs). Design/methodology/approach In fact, TO problems are known to be large-scale optimization problems. The authors therefore adopt the adjoint method to reduce the computation time required to obtain the gradient information. In this paper, they illustrate the continuous variant of the adjoint method in the context of magnetostatics TO. Herein, the authors propose an implementation of the adjoint method then use it within a gradient-based optimization solver fmincon-MATLAB to solve a HET TO design problem. Findings By comparison with finite difference method, the authors validate the accuracy of the suggested implementation of the adjoint method. Then, they solve a large-scale HET TO design problem. The resultant design of TO is distinctly original and not intuitive. Research limitations/implications In this paper, the authors introduce TO as a tool that has allowed them to explore new and innovative design of a HET. However, although the design presented is original, its manufacture is not feasible. Thus, a discussion section has been included at the end of paper to suggest a possible way to concretize topological solutions. Practical implications TO helps to explore more original design possibilities. In this paper, the authors present an implementation of the adjoint method that makes it possible to solve efficiently and in less central processing unit time large-scale TO design problem. Originality/value An easy implementation of the adjoint method is presented in magnetostatics TO. This implementation was first validated by comparison with the finite difference method and then used to solve a large-scale design problem. The result of the TO design problem is distinctly original and non-intuitive.

Adjoint-based topology optimization to reduce the energy consumption of a Hall effect thruster

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rtimi Youness ◽  
Frederic Messine
Keyword(s):  
Topology Optimization ◽  
Energy Consumption ◽  
Hall Effect ◽  
Design Problem ◽  
Design Methodology ◽  
Large Scale ◽  
Material Distribution ◽  
Original Design ◽  
Content Type ◽  
Hall Effect Thruster

Purpose The presented study aims to minimize the energy consumed by a Hall effect thruster (HET) under a constraint which makes it possible to generate a specified magnetic field in a target region of the thruster. Design/methodology/approach Herein topology optimization (TO) is used to reduce the energy consumption of an HET while keeping its performance unchanged. The design variables are the current densities in the coils and the distribution of materials in the polar pieces of the thruster. Intermediate values of material distribution are penalized using the solid isotropic material with penalization method to favor binary solutions. By means of the adjoint method, this paper provides the derivatives of the objective and constraint functions with respect to material distribution and current density variables. Findings The TO-based design methodology is developed and validated on a design example involving 2,051 variables. The approach shows its interest and its effectiveness of on a large scale two-criteria problem. Research limitations/implications In this paper, TO is presented as a tool that has allowed to explore new and innovative designs. However, although the design presented is original, its fabrication is not feasible. Despite this, the designs found give a good idea of the starting points for shape and parametric optimization tools. Practical implications Through the HET design problem, TO shows the ability to explore more original design possibilities of a complex magnetostatic design problem and to discover designs that make a HET more efficient with respect to several criteria at the same time. Originality/value A new way to reduce the energy consumption of a HET is presented. To achieve this, an adjoint-based TO method is developed and then implemented in a simple way. This approach shows that, for efficiency purposes, TO is a key tool for extending the state of the art of HET designs.

Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part I: Methodology and Verification

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
D. X. Wang ◽  
L. He
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Design Optimization ◽  
Difference Method ◽  
Method Development ◽  
Adjoint Method ◽  
Flow Characteristics ◽  
Compressor Stage ◽  
Flow Domain ◽  
Domain Conservation

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multistage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. Attention is paid to the exactly opposite propagation of the adjoint characteristics against the physical flow characteristics, providing a simple and consistent guidance in the adjoint method development and applications. The adjoint mixing-plane treatment is formulated to have the two fundamental features of its counterpart in the physical flow domain: conservation and nonreflectiveness across the interface. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment.

A Quasi-First Order Optimization Method and its Demonstration on the Optimization of a U-Bend

2015 ◽  
Author(s):  
M. Bugra Akin ◽  
Wolfgang Sanz
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Adjoint Method ◽  
Optimization Method ◽  
Second Order ◽  
Computational Time ◽  
First Order ◽  
Finite Differences Method ◽  
The Finite Difference Method

Optimal shape design is widely used today to improve a variety of designs. It is a challenging task and several methods have been developed. These methods are generally classified by the order of derivatives used. They are zero, first and second order methods, which, as their names imply, use only the function values, first and second order derivatives, respectively. There are two common approaches to first order methods. These are the finite difference method and the adjoint method. The finite difference method requires an additional CFD calculation for each parameter, which quickly becomes computationally very expensive as the number of parameters rise. The adjoint method provides a computationally efficient alternative in such cases. But the computational cost of the adjoint method also becomes expensive if additional constraints are introduced or when multi-objective optimizations are considered. This paper presents a novel optimization strategy which can be classified as a quasi-gradient based optimization method. As with the finite differences method an additional CFD calculation is performed for each parameter. But in order to save computational time the simulations are not performed to full convergence so that the derivatives are not calculated accurately. The only information that can be obtained in this way is whether the chosen contour manipulation leads to an improvement. A line search method is introduced that can find an optimum using this incomplete gradient information. The optimization method is demonstrated by the quasi-3d optimization of a U-bend.

Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part I—Methodology and Verification

2008 ◽  
Author(s):  
D. X. Wang ◽  
L. He
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Design Optimization ◽  
Difference Method ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Steepest Descent Method ◽  
Compressor Stage ◽  
Multi Stage ◽  
Continuous Adjoint

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multi-stage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. The gradients of a single objective function of a weighted sum of objectives and constraints with respect to detailed blade shape perturbations are obtained very efficiently by the continuous adjoint method. The steepest descent method is used to drive the design to an optimum. The adjoint mixing-plane treatment enables the adjoint equations to be solved in a multi-stage environment. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment.

A gradient-based shape optimization scheme via isogeometric exact reanalysis

2018 ◽  
Vol 35 (8) ◽  
pp. 2696-2721 ◽  
Author(s):  
Chensen Ding ◽  
Xiangyang Cui ◽  
Guanxin Huang ◽  
Guangyao Li ◽  
K.K. Tamma ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Shape Optimization ◽  
Difference Method ◽  
Computer Aided Design ◽  
Content Type ◽  
Optimization Framework ◽  
Computer Aided ◽  
Gradient Based ◽  
The Finite Difference Method

PurposeThis paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.Design/methodology/approachThe authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.FindingsThe proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.Originality/valueThe authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.

Natural convection inside a C-shaped nanofluid-filled enclosure with localized heat sources

2014 ◽  
Vol 24 (8) ◽  
pp. 1954-1978 ◽  
Author(s):  
M.A. Mansour ◽  
M.A. Bakeir ◽  
A. Chamkha
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Finite Difference ◽  
Aspect Ratio ◽  
Finite Difference Method ◽  
Difference Method ◽  
Volume Fraction ◽  
Cu Nanoparticles ◽  
Content Type ◽  
The Finite Difference Method

Purpose – The purpose of this paper is to investigate natural convection fluid flow and heat transfer inside C-shaped enclosures filled with Cu-Water nanofluid numerically using the finite difference method. Design/methodology/approach – In this investigation, the finite difference method is employed to solve the governing equations with the boundary conditions. Central difference quotients were used to approximate the second derivatives in both the X and Y directions. Then, the obtained discretized equations are solved using a Gauss-Seidel iteration technique. Findings – It was found from the obtained results that the mean Nusselt number increased with increase in Rayleigh number and volume fraction of Cu nanoparticles regardless aspect ratio of the enclosure. Moreover the obtained results showed that the rate of heat transfer increased with decreasing the aspect ratio of the cavity. Also, it was found that the rate of heat transfer increased with increase in nanoparticles volume fraction. Also at low Rayleigh numbers, the effect of Cu nanoparticles on enhancement of heat transfer for narrow enclosures was more than that for wide enclosures. Originality/value – This paper is relatively original for considering C-shaped cavity with nanofluids.

scholarly journals Modelling and simulation of heat conduction in 1-D polar spherical coordinates using control volume-based finite difference method

2016 ◽  
Vol 26 (1) ◽  
pp. 2-17 ◽  
Author(s):  
Attila Diószegi ◽  
Éva Diószegi ◽  
Judit Tóth ◽  
József Tamás Svidró
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Coordinate System ◽  
Heat Transport ◽  
Difference Method ◽  
Control Volume ◽  
Polar Coordinate System ◽  
Heat Absorption ◽  
Content Type ◽  
Polar Coordinate

Purpose – The purpose of this paper is to obtain a finite difference method (FDM) solution using control volume for heat transport by conduction and the heat absorption by the enthalpy model in the sand mixture used in casting manufacturing processes. A mixture of sand and different chemicals (binders) is used as moulding materials in the casting processes. The presence of various compounds in the system improve the complexity of the heat transport due to the heat absorption as the binders are decomposing and transformed into gaseous products due to significant heat shock. Design/methodology/approach – The geometrical domain were defined in a 1D polar coordinate system and adapted for numerical simulation according to the control volume-based FDM. The simulation results were validated by comparison to the temperature measurements under laboratory conditions as the sand mould mixture was heated by interacting with a liquid alloy. Findings – Results of validation and simulation methods were about high correspondence, the numerical method presented in this paper is accurate and has significant potential in the simulation of casting processes. Originality/value – Both numerical solution (definition of geometrical domain in 1D polar coordinate system) and verification method presented in this paper are state-of-the-art in their kinds and present high scientific value especially regarding to the topic of numerical modelling of heat flow and foundry technology.

Parallel Algorithms for Compressible Turbulent Flow Simulation Using Direct Numerical Method

2012 ◽  
Vol 516-517 ◽  
pp. 980-991
Author(s):  
De Bo Li ◽  
Qi Sheng Xu ◽  
Yue Liang Shen ◽  
Zhi Yong Wen ◽  
Ya Ming Liu
Keyword(s):  
Parallel Computing ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Large Scale ◽  
Two Phase ◽  
Direct Numerical Method ◽  
Explicit Finite Difference ◽  
Speedup Factor ◽  
Explicit Finite Difference Method

In this study, SPMD parallel computation of compressible turbulent jet flow with an explicit finite difference method by direct numerical method is performed on the IBM Linux Cluster. The conservation equations, boundary conditions including NSCBC (charactering boundary conditions), grid generation method, and the solving processing are carefully presented in order to give other researchers a clear understanding of the large scale parallel computing of compressible turbulent flows using explicit finite difference method, which is scarce in the literatures. The speedup factor and parallel computational efficiency are presented with different domain decomposition methods. In order to use our explicit finite method for large scale parallel computing, the grid size imposed on each processor, the speedup factor, and the efficiency factor should be carefully chosen in order to design an efficient parallel code. Our newly developed parallel code is quite efficient from that of implicit finite difference method or spectral method on parallel computational efficiency. This is quite useful for future research for gas and particle two-phase flow, which is still a problem for highly efficient code for two-phase parallel computing.

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