Continuum Design Sensitivity Research Articles

Continuum sensitivity and design optimization of superconducting systems under critical current densities with magnetic field dependence

This paper presents an approach for deriving the continuum sensitivity of superconducting systems operating at critical current densities and an optimization method based on the continuum sensitivity. In the sensitivity problem, the superconducting systems is represented by a variational state equation, wherein the magnetic permeability depends on the magnetic field, which is transformed from a state equation with a field-dependent source. The design sensitivity is derived using the material derivative concept of continuum mechanics and the adjoint variable method. The adjoint system has a material property represented as a symmetric tensor that contains the sensitivity of the current density with respect to the magnetic field. The design sensitivity is represented in the analytical form of a surface integral on the interface between the superconducting material and its surroundings, which depends on the sensitivity of the current density. The optimization scheme is constructed based on the continuum design sensitivity. In the design optimization, the level set method is used to express the shape variation of the superconducting materials. The numerical example of infinite solenoids demonstrates that the design sensitivity provides an accurate design solution considering the critical current condition. In addition, the design example of a magnetic resonance imaging solenoid shows that the derived design sensitivity has the inherent ability for attaining the compact design by treating the input current of a superconducting system as a critical condition.

Alleviation of Electric Field Intensity in High-Voltage System by Topology and Shape Optimization of Dielectric Material Using Continuum Design Sensitivity and Level Set Method

A new design method for topology and shape optimization to preserve high-voltage systems is proposed. The level set technique for the topology optimization is employed with the finite-element method. For the velocity field in the level set equation, the continuum sensitivity formula is used for accurate information of the electric field effect on the topology and shape variation. This technique is applied to three practical examples and the optimum designs of insulating dielectric for the systems are obtained. The optimum designs could suppress troubles, such as thermal deterioration or electrical breakdown, in the systems.

Topology optimization of rotor poles in a permanent-magnet machine using level set method and continuum design sensitivity analysis

Purpose– In this paper, a numerical approach to the topology optimization is proposed to design the permanent magnet excited machines with improved high-speed features. For this purpose the modified multi-level set method (MLSM) was proposed and applied to capture the shape of rotor poles on the fixed mesh using FE analysis. The paper aims to discuss these issues.Design/methodology/approach– This framework is based on theories of topological and shape derivative for the magnetostatic system. During the iterative optimization process, the shape of rotor poles and its evolution is represented by the level sets of a continuous level set function f. The shape optimization of the iron and the magnet rotor poles is provided by the combining continuum design sensitivity analysis with level set method.Findings– To obtain an innovative design of the rotor poles composed of different materials, the modified MLSM is proposed. An essential advantage of the proposed method is its ability to handle a topology change on a fixed mesh by the nucleating a small hole in design domain that leads to more efficient computational scheme then standard level set method.Research limitations/implications– The proposed numerical approach to the topology design of the 3D model of a PM machine is based on the simplified 2D model under assumption that the eddy currents in both the magnet and iron parts are neglected.Originality/value– The novel aspect of the proposed method is the incorporation of the Total Variation regularization in the MLSM, which distribution is additionally modified by the gradient derivative information, in order to stabilize the optimization process and penalize oscillations without smoothing edges.

Optimal Design of a RFID Tag Antenna Based on Plane-Wave Incidence

In order to increase the detection range and radar cross section (RCS), the patch shape of a UHF RFID tag antenna is optimized using continuum design sensitivity analysis. To maximize RCS and reflect the practical RFID tag operating condition, the design goal is set considering not only the matching characteristics of the antenna but also the induced voltage at the RFID chip terminal when the plane-wave is incident upon the antenna. The complex impedance of the RFID chip is taken into account by defining a lumped port at the chip location. The design results show that the induced voltage at the chip has increased more than 32% after the optimization.

Generalized Continuum Sensitivity Formula for Shape Optimization of High-Frequency Devices in Frequency Domain

This paper presents a generalized continuum design sensitivity formula that is applicable to any shape design problems of high-frequency devices in frequency domain. The derived sensitivity formula is expressed in terms of surface integrals over a part of the interface between different materials and of the conducting boundary undergoing shape modification during iterative design process. Using the proposed approach, it is easy to deal with the shape optimization concerning conducting surfaces as well as dielectric contours. The method has been successfully applied to the optimization of high-frequency waveguides where the numerical implementation is thoroughly investigated from the viewpoint of constructing adjoint systems and calculating sensitivity values.

Design Optimization of Waveguide Filters Using Continuum Design Sensitivity Analysis

This paper presents a new methodology for design optimization of dielectric waveguide filters based on the continuum design sensitivity analysis in conjunction with standard electromagnetic analysis codes. To achieve this, first, an analytical sensitivity formula in the frequency domain is systematically derived by exploiting the augmented Lagrangian method, material derivative concept and adjoint variable method. Then unified program architecture integrating several engineering software packages into a design tool is proposed for optimum design of high-frequency devices. A 3-D dielectric resonator used in waveguide filters has been tested to prove the validity of the proposed method.

Optimization Design of a Dielectric Lowpass Filter based on Continuum Design Sensitivity Analysis in Frequency Domain

This paper presents a new methodology for designing a dielectric waveguide filter with the cutoff frequency of 2.4 GHz based on the continuum design sensitivity analysis. An analytical sensitivity formula is derived in frequency domain and then unified program architecture applicable to the optimal design of high-frequency devices is proposed. A three-dimensional dielectric resonator used in waveguide filters has been tested to illustrate the validity of the proposed method.

Local Electromagnetic Force Computation in the Presence of Numerical Field Errors

Several approaches exist for the evaluation of force distributions in electromagnetic devices. The local virtual work based stresses derived from the application of continuum design sensitivity analysis to the variation of system energy are equivalent to the traditional Maxwell stress tensor formulation in total force calculation. While both methods rely on local field values, they produce different force distributions for the same field. The paper reviews these results and examines the discrepancy due to the numerical errors in the field solution and the difference in the permeability of the material across the interface.

General Formulation for Magnetic Forces in Linear Materials and Permanent Magnets

A previous work described how a magneto-mechanical tensor can be obtained from the application of the virtual work principle to the energy density or co-energy density in any given media. This paper describes how the volume and surface force densities can be obtained from this tensor. Taking these force densities into account, we can calculate the total force over any given magnetic medium that is surrounded by any other magnetic medium. This represents a generalization of the classic Maxwell theorem. With this methodology, it is possible to calculate the surface forces in bodies in contact. We demonstrate this by showing the equivalence of the results obtained by this procedure and the equivalent source method and the Kelvin expression method. These methods could previously be applied only to bodies surrounded by air. We also show that this procedure is equivalent to other methods used to calculate the forces between linear magnetic materials (Korteweg-Helmholtz's tensor, ponderomotive force density). We have compared our method with results from continuum design sensitivity analysis, a method that allows for the calculation of global and local forces in magnetic media in contact .

A Sensitivity Approach to Force Calculation in Electrostatic MEMS Devices

This paper presents a novel method for the computation of force in an electrostatically operated microelectromechanical systems (MEMS) device. The approach is based on continuum design sensitivity analysis (CDSA) and can be used with any analysis system. The method, unlike that of Maxwell stresses, does not require an airgap surrounding the body. The method is applied to the calculation of the tilt angle of a MEMS micromirror, and numerical results are compared with experimental measurements on the real device.

Smooth boundary topology optimisation applied to an electrostatic actuator

Smooth boundary topology optimisation in conjunction with the continuum design sensitivity analysis avoids many of the problems encountered by conventional cell-based systems coupled with material homogenisation or the density method. Shape optimisation becomes part of topology optimisation. The effectiveness of the proposed method is demonstrated through the design of an electrostatic MEMS actuator to generate maximum torque for a predefined maximum size (area).

Efficient Global and Local Force Calculations Based on Continuum Sensitivity Analysis

Equations derived from the continuum design sensitivity analysis (CDSA), in conjunction with the material derivatives for a continuous medium and using the energy-based approach, have been successfully applied to the calculation of both total force and force distributions. The resultant expressions are similar to the Maxwell Stress Tensor, Magnetic Charge Method, and Virtual Work Method but have several advantages over the traditional approaches. Numerical implementation of the scheme leads to efficient calculations and improved accuracy

Design optimisation of electromagnetic devices using continuum design sensitivity analysis combined with commercial EM software

The paper deals with two types of optimisation problem: optimised source distribution and the shape optimum design, using continuum design sensitivity analysis (CDSA) in combination with standard electromagnetic (EM) software. Fast convergence and compatibility with existing EM software are the distinctive features of the proposed implementation. To verify the advantages and also to facilitate understanding of the method itself, two design optimisation problems have been tested using both 2D and 3D models: the first is an MRI design problem related to finding an optimum permanent magnet distribution, and the second is a pole shape design problem to reduce the cogging torque in a BLDC.

Efficient force calculations based on continuum sensitivity analysis

Using continuum design sensitivity analysis (CDSA), in conjunction with the virtual work principle, equations have been derived for calculating forces without the need to solve the adjoint system. The resultant expressions are similar to the Maxwell stress tensor, but have the important advantage of the integration taking place on the surface of material rather than in the air outside. Implementation of the scheme leads to efficient calculations and improved accuracy.

Topology optimization of electromagnetic systems considering magnetization direction

The topology optimization of electromagnetic (EM) systems considering the effect of the magnetization direction in the magnet is investigated using the finite element method. The density method is used for topology optimization and continuum design sensitivity analysis is used for the sensitivity of EM systems. Also ANSYS is used for analysis. The force of a magnet is divided into two components of x and y direction in the rectangular coordinates. And the topology optimization of a c-core and a brushless dc motor are obtained.

Numerical investigation of the electric field distribution induced in the brain by transcranial magnetic stimulation

Results are presented on the prediction and optimisation of the electric field distribution obtained during transcranial magnetic stimulation (TMS) for deep neuron stimulation by using the finite-element method (FEM) in three dimensions. The effects of the geometrical models of the head on the distribution and penetration of the electric field induced in the brain during TMS are examined. For a magnetic field that can penetrate deeply and safely to activate the brain's central structures, an iron core is introduced and its core shape is optimised using continuum design sensitivity analysis (CDSA) combined with the FEM. It is revealed that the incorporation of an accurate brain model in terms of shape as well as conductivity values is crucial for improved estimation of the field distribution. The introduction of an optimised iron core is shown to enhance the magnitude and localisation of the electric field induced inside the brain.

Applying Continuum Design Sensitivity Analysis Combined With Standard EM Software to Shape Optimization in Magnetostatic Problems

This paper explores the physical meaning of pseudosources of an adjoint system in a continuum design sensitivity analysis (CDSA) when applied to shape optimization in magnetostatics. An efficient and practical way to compute the required gradient information with standard electromagnetic (EM) software packages is suggested. Based on this novel methodology, designers will be able to deal with practical design optimization of electro-mechanical devices using existing analysis tools without the need to access the complicated code of the EM software. The applicability of the proposed methodology is demonstrated by optimizing the salient pole face shape of a high-temperature superconducting synchronous generator.

Parametric design sensitivity analysis of high-frequency structural-acoustic problems using energy finite element method

A design sensitivity analysis of high-frequency structural-acoustic problems is formulated and presented. The energy finite element method (EFEM) is used to predict structural-acoustic responses in the high frequency range, where the coupling between structural junctions and the structural-acoustic interface are modelled using power transfer coefficients. The continuum design sensitivity formulation is derived from the governing equation of EFEM and the discrete method is applied in the variation of the structural-structural and structural-acoustic coupling matrices. The direct differentiation and adjoint variable method are both developed for the sensitivity analysis, where the difficulty of the adjoint variable method is overcome by solving a transposed system equation. Parametric design variables such as panel thickness and material damping are considered for sensitivity analysis, and numerical sensitivity results show excellent agreement as compared to analytical finite difference results.

Configuration Design Sensitivity Analysis for Dynamic Systems Using CAD-Based Velocity Field

Nomenclature aA.•; •/ = bilinear form due to strain energy a 0 Vμ .•; •/ = variation of aA.•; •/ due to orientation change a 0 VA .•; •/ = variation of aA.•; •/ due to shape change cA.•; •/ = bilinear form due to the damping dA.•; •/ = bilinear form due to kinetic energy e.•; •/ = bilinearmapping h = thickness of the design component `A.•/ = load linear form V = design velocity Ž eld Vμ = orientation design velocity Ž eld VA = shape design velocity Ž eld y = eigenvector of structure Z = space of kinematically admissible displacements z = displacementvector 1A = sensitivity coefŽ cient from central Ž nite differencemethod 3 = eigenvalue of structure 3 0 = variation of 3 due to design change 3 0 Vμ = variation of 3 due to orientation change 3 0 VA = variation of 3 due to shape change = adjoint displacement vector 1⁄2 = mass density & = hysteretic damping coefŽ cient A 0 = continuumdesign sensitivity coefŽ cient ! = natural frequency of structure r = gradient operator

Design Sensitivity Analysis of Noise, Vibration, and Harshness of Vehicle Body Structure*

ABSTRACT A continuum design sensitivity analysis (DSA) approach to noise, vibration, and harshness (NVH) of vehicle body structure is developed using Mindlin plate and Timoshenko beam theories. Linear shape functions used for numerical meth ods of DSA cause a deficiency in bending energy that is accounted for. For large scale NVH applications, linear and bilinear isoparametric elements require much less analysis time. To handle complex vehicle built-up structures, the warped plate element and general cross-sectional beam element with offset effects are included in the numerical method for DSA. A variational approach is used to obtain the continuum design sensitivity formulation. Thickness of plates and nonprincipal section parameters of beams are selected as design variables. A numerical implementation method for continuum DSA is developed by postprocessing analysis results from established finite element analysis (FEA) codes to obtain NVH design sensitivity. The numerical method is tested using large-scale concept and detailed passenger vehicle models. Design sensitivity results are validated using reanalysis results.