Continuum Design Sensitivity Analysis Research Articles

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.

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.

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.

Design Sensitivity Analysis of Structure-Induced Noise and Vibration

A continuum design sensitivity analysis (DSA) method for dynamic frequency responses of structural-acoustic systems is developed using the adjoint variable and direct differentiation methods. A variational approach with a non-self-adjoint operator for complex variables is used to retain the continuum elasticity formulation throughout derivation of design sensitivity results. It is shown that the adjoint variable method is applicable to the variational equation with the non-self-adjoint operator. Sizing design variables such as the thickness and cross-sectional area of structural components are considered for the design sensitivity analysis. A numerical implementation method of continuum DSA results is developed by postprocessing analysis results from established finite element analysis (FEA) codes to obtain the design sensitivity of noise and vibration performance measures of the structural-acoustic systems. The numerical DSA method presented in this paper is limited to FEA and boundary element analysis (BEA) is not considered. A numerical method is developed to compute design sensitivity of direct and modal frequency FEA results. For the modal frequency FEA method, the numerical DSA method provides design sensitivity very efficiently without requiring design sensitivities of eigenvectors. The numerical method has been tested using passenger vehicle problems. Accurate design sensitivity results are obtained for analysis results obtained from established FEA codes.

Shape design sensitivity analysis and optimization of spatially rotating objects

Shape design sensitivity analysis (DSA) and optimization of spatially rotating objects is presented in this paper. Design sensitivity expressions are derived using a continuum DSA method for spatial objects rotating with angular velocity and angular acceleration, based on three definitions of the finite element mass matrix: consistent, lumped, and diagonalized. The design sensitivity expression derived using a diagonalized element mass matrix, which is consistent with the finite element analysis (FEA) method used in ANSYS, is implemented, although the method can work with other FEA codes, such as MSC/NASTRAN or ABAQUS. Since the continuum DSA method is used, sensitivity information can be computed outside the FEA codes by postprocessing finite element data. Rotating block and turbine blade examples are presented to validate the proposed DSA method. The turbine blade example is optimized using an integrated optimization module of the Design Sensitivity Analysis and Optimization (DSO) tool developed at the University of Iowa. The integrated module consists of ANSYS, MSC/NASTRAN, or ABAQUS for FEA; Design Optimization Tool (DOT) for nonlinear programming; and DSA and design model update programs developed in DSO.

Design Sensitivity Analysis and Optimization of Nonlinear Structural Systems With Critical Loads

A continuum formulation for design sensitivity analysis of critical loads is developed for nonlinear structural systems that are subjected to conservative loading. Both geometric and material nonlinear effects are considered. Sizing design variables such as cross-sectional areas of beam or truss design components and thicknesses of plate or membrane design components, together with their shape design variables, are treated. A continuum approach is used to obtain design sensitivity expressions in integral form. For shape design sensitivity analysis, the material derivative concept and domain method are used to find variations of the critical load due to variations in shape of the physical domain. The total Lagrangian formulation for incremental equilibrium equation and one-point linearized eigenvalue problems are utilized. A numerical method is presented to evaluate continuum design sensitivity expressions using analysis results of established finite element codes. It is found that no adjoint system is necessary for design sensitivity analysis of the critical load. Numerical results show the proposed method for design sensitivity of critical loads is accurate for both sizing and shape design variables. A numerical procedure for optimal design of nonlinear structural systems is presented, using the proposed continuum design sensitivity analysis method. An optimal design problem with a stability constraint is solved.

Sizing Design Sensitivity Analysis of Dynamic Frequency Response of Vibrating Structures

A continuum design sensitivity analysis method of dynamic frequency response of structural systems is developed using the adjoint variable and direct differentiation methods. A variational approach with a non-selfadjoint operator for complex variable is used to retain the continuum elasticity formulation throughout derivation of design sensitivity results. Sizing design variables such as thickness and cross-sectional area of structural components are considered for the design sensitivity analysis. A numerical implementation method of continuum design sensitivity analysis results is developed using postprocessing analysis data of COSMIC/NASTRAN finite element code to get the design sensitivity information of displacement and stress performance measures of the structures. The numerical method is tested using basic structural component such as a plate supported by shock absorbers and a vehicle chassis frame structure for sizing design variables. Accurate design sensitivity results are obtained even in the vicinity of resonance.

An interactive post-processor for structural design sensitivity analysis and optimization: Sensitivity display and what-if study

A menu-driven, multi-windowing design workstation for interactive design optimization is presented. As an alternative to a black-box optimization approach, the design workstation supports a four-step interactive design methodology that consists of (1) displaying design sensitivity information; (2) carrying out what-if studies; (3) determining trade-offs; and (4) performing interactive design optimization. Graphics capabilities of established finite element pre- and post-processors are exploited in providing for visualization of results generated during the interactive design process, within a graphics workstation environment. By using recently-developed methods of continuum design sensitivity analysis and a local database, established finite element analysis codes can be used for structural analysis and adjoint load reanalysis.