Design sensitivity analysis for shape optimization based on the Lie derivative

peer reviewedAbstract The paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations. The proposed approach, based on geometrical concepts borrowed from differential geometry, shows that sensitivity of a performance function (i.e. any function of the solution of the problem) with respect to a given design variable can be represented mathematically as a Lie derivative, i.e. the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable. Theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper, and applied to a nonlinear magnetostatic and a linear elastic problem, following both the direct and the adjoint approaches. Following the analytical approach, one linear system of which only the right-hand side needs be evaluated (the system matrix being known already) has to be solved for each of the design variables in the direct approach, or for each performance functions in the adjoint approach. A substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity, which requires solving a second nonlinear system for each design variable. This is the main motivation of the analytical approach. There is some freedom in the definition of the auxiliary flow that represents the shape modification. We present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification. All sensitivity calculations are checked with a finite difference in order to validate the analytic approach. Convergence is analyzed in 2D and 3D, with first and second order finite elements

Remake of a Magento web store content management system into system for employees\u27 attendance tracking

Diplomska naloga obsega predelavo Magento sistema za upravljanje vsebin spletne trgovine v sistem za evidenco dela zaposlenih. V prvem delu diplomskega dela so najprej predstavljeni razlogi za izbor Magento sistema, prikazane pa so tudi nekatere ze obstojece spletne resitve za evidenco dela zaposlenih. Sledi predstavitev Magento sistema, oziroma princip delovanja njegove MVC arhitekture. Nato sledi opis orodij, s katerimi je izdelan Magento sistem, ter opis orodij, ki smo jih dodali za ucinkovitejse in uporabniku prijaznejse delovanje nasega sistema za evidenco dela. V drugem delu je opisan postopek izdelave, ki se zacne s postopkom redukcije Magento sistema, v bolj splosen sistem za razvoj spletnih aplika- cij. S pomocjo primera implementacije glavnega modula, ki sluzi sistemu za evidenco dela zaposlenih, prikazemo razvoj modulov v Magento sistemu in predstavimo tudi primer razsiritve obstojece funkcionalnosti Magento sis- tema. Sledi se opis vseh ostalih funckionalnosti, ki smo jih implementirali za sistem za evidenco dela, ter opis izdelave dizajna, ki poskusa slediti najno- vejsim trendom pri razvoju spletnih aplikacij. V zadnjem delu pa so predstavljeni cilji nadaljnega razvoja aplikacije, kjer nekateri temeljijo na ze obstojecih funkcionalnostih Magento sistema.This thesis comprises a rework of the Magento e-commerce content manage- ment system into an employee attendance tracking system. In the rst part of the thesis the reasons for selecting Magento system are presented and also some already existing web solutions for employee attendance tracking are shown. Next is the presentation of the Magento system, that is, principles of its MVC architecture. After that follows the description of the tools on which Magento is based upon and some of the tools we used to make our employee attendance tracking software better. The second part describes the development process, which starts with the reduction of the Magento system, to a more general system for web based application development. With the help of the implementation process of our main module, which serves the purpose of the employee attendance tracking system, we describe the module development process in Magento system and present an example of extending a core functionality. Next, the presentation of the remaining functionalities implemented for our employee attendance tracking system follows, and the description of the design, which attempts to follow the newest trends in web application development. In the last part we present the goals of the future development of the application, where some of them are based on already existing functionalities of the Magento system

Shape and Topology Optimization for electro-mechanical energy converters

The sustained growth of the industrial sector requires high-efficiency electro-me- chanical energy converters, in particular electrical rotating machines, at the lowest possible cost. The use of modern power electronics converters at all levels of elec- trical power applications, involves, on the other hand, switching components with very low switching times and always increasing current levels. Passive components in these devices (busbars, inductors, transformers) must be designed to be com- pact without compromising their performance (e.g. power losses, electromagnetic interference/compatibility). Automated design optimization methods, in particu- lar shape and topology optimization, used so far mostly in the field of structural engineering, offer a major step evolution in the design of such electro-mechanical and electric energy converters. The objective of this thesis is to provide engineers and practitioners of the field with appropriate methods which allow to carry out such design tasks by numerical optimization in an efficient way, and to extend the design capabilities to electro-mechanical converters. This thesis exploits a computer aided design (CAD) representation of indus- trial systems and the finite element method (FEM) to solve the partial differential equations (PDEs) that govern their behavior under certain physical conditions. This thesis addresses three main subjects. First, the sensitivity analysis of elec- tromagnetic PDEs solution is revisited in view of being used with gradient-based methods. Classical scalar formulations are extended to a general rigorous frame- work, and expressed analytically prior to discretization, to treat the vector case. Secondly, an iterative solver is designed so as to solve efficiently the large-scale linear systems arising from the design problem. Third, the design improvement capabilities are extended by developing an integrated and unified formalism for simultaneous shape and topology optimization of a system

Design Sensitivity Analysis for Shape Optimization of Nonlinear Magnetostatic Systems

In this paper, a direct and an adjoint analytic sensitivity analysis for a nonlinear magnetostatic system is obtained, in the context of shape optimization for any design function. The calculations are based on the material derivative concept of continuum mechanics. The resulting sensitivity formula can be expressed as either a volume integral or as a boundary integral along the interface where shape modification occurs. A method for the calculation of the design velocity field and mesh updating scheme is introduced as well. The accuracy of the methodology is analysed on an inductor system, suggesting that the volume integration technique should be preferred. All methods are freely available for further testing in the open source environment GetDP/Gmsh

Shape and Topology Optimization of Electrical Machines using Lie Derivative-Based Analytical Sensitivity Analysis

peer reviewedThe paper addresses the optimal design of electric machines, through the general setting of both shape and topology optimization. The optimization problems are efficiently solved with a classical gradient-based mathematical programming algorithm. An analytical sensitivity analysis for the nonlinear magnetostatic problem that can handle both shape and topology design variables, based on the Lie derivative is derived and applied to the optimal design of an interior permanent magnet (IPM) machine

Three-dimensional Topology Optimization of Planar Multilayer Busbar

This paper addresses the optimal design of three- dimensional planar multilayer busbar, through the general setting of topology optimization. The optimization problem is efficiently solved with a gradient-based mathematical program- ming algorithm, that exploits a harmonic adjoint variational formulation sensitivity analysis for the harmonic linear mag- netodynamic problem. This formulation can handle topology design variables defined on a finite element mes

Ocena vodne bilance za lizimetrsko postajo pri Vodarni Kleče v Ljubljani

The aim of shape optimization is to determine the layout of the structure that minimizes a performance measure, under some design constraints. Mathematical programming, which relies on the derivative of all the performance measures, is adopted to solve the problem. Thus, it is crucial to establish an efficient and accurate method for the calculation of the sensitivity of a response to shape variation

Shape and topology optimization of electrical machines using lie derivative-based analytical sensitivity analysis

The paper addresses the optimal design of electric machines, through the general setting of both shape and topology optimization. The optimization problems are efficiently solved with a classical gradient-based mathematical programming algorithm. An analytical sensitivity analysis for the nonlinear magnetostatic problem that can handle both shape and topology design variables, based on the Lie derivative is derived and applied to the optimal design of an interior permanent magnet (IPM) machin