### Abstract

intensified. The increase in computational power, and the emergence of new optimisation algorithms provided new instruments for the design of magnetic systems. All these factor contribute in making the optimisation of magnetic systems a very lively sector of modern research.

The main focus of this work are magnetic systems based on permanent magnets, although hybrid systems combining permanent magnets with electromagnets are also considered. Many optimisation approaches presented here are derived within a framework based on the reciprocity theorem. This theorem formulates an energy equivalence principle with several implications concerning the optimisation of objective functionals that are linear with respect to the magnetic field. Linear functionals represent different optimisation goals, e.g. maximising a certain component of the field averaged over a region of space. In general, a linear functional can be expressed as the integral over a given region of the scalar product between the magnetic field and an arbitrarily defined objective vector field. It has been known for some time that the reciprocity theorem can be used to determine the optimal remanence distribution with respect to a linear objective functional.

Additionally, it is shown here that the same formalism can be applied to the optimisation of the geometry of magnetic systems. Specifically, the border separating the permanent magnet from regions occupied by air or soft magnetic material can be optimised within this framework. Since in the practice most structures are realized by assembling uniformly magnetized pieces of permanent magnet, it is relevant to address the question of how a given region of space is best subdivided. This problem is investigated here within the framework of the reciprocity theorem. Analytical derivations will be used to show that, for segmentations controlled by a single parameter, the globally optimal solution to this problem can be determined for almost arbitrary geometries. The case of segmentations depending by two parameters has been approached employing a heuristic algorithm, which led to new design concepts. Some of the procedures developed for linear objective functionals have been extended to non-linear objectives, by employing iterative techniques.

Even though most the optimality results discussed in this work have been derived analytically, the different approaches have been implemented in combination with finite element methods, resulting in

flexible and computationally efficient algorithms. Most of the optimisation approaches could only be proven under the assumption of linear magnetic behavior. The last part of this thesis also investigates some of the effects on the performance of magnetic systems, due to non-linear magnetic phenomena. In particular, the non-linear demagnetization effects caused by the finite coercivity of the permanent magnet material will be examined.

All the optmisation techniques will be illustrated with example magnetic systems for different applications, thus showing the versatility and efficacy of the various approaches. The Halbach cylinder geometry, relevant for many applications, will be often used as example, also because of the many symmetries and optimality properties exhibited by this geometry. Despite the fact that this system has already been subject of many publications, some of the aspects considered in this thesis have not been investigated before. The ultimate goal of the PhD project is to apply the optimisation techniques developed during this research to the design of the magnetic system for the prototype of heat pump based on the magnetocaloric effect. Magnetic systems for room temperature magnetic refrigeration will thus frequently be used as illustrative examples along the course of this thesis.

Primarily because of the theoretical relevance of linear functionals, the results presented here lead to a deeper understanding of the magnet optimisation process. One of the perspectives considered in this work is the trade-off between field intensity and field quality, as the choice of a particular optimisation approach may favour one or the other. The general framework discussed here provides a set of useful tools aiding the magnet design process. This research also opened new scientific questions which would be worth investigating in future studies.

Original language | English |
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Publisher | Department of Energy Conversion and Storage, Technical University of Denmark |
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Number of pages | 236 |

Publication status | Published - 2016 |

### Cite this

*Optimising Magnetostatic Assemblies*. Department of Energy Conversion and Storage, Technical University of Denmark.

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*Optimising Magnetostatic Assemblies*. Department of Energy Conversion and Storage, Technical University of Denmark.

**Optimising Magnetostatic Assemblies.** / Insinga, Andrea Roberto.

Research output: Book/Report › Ph.D. thesis

TY - BOOK

T1 - Optimising Magnetostatic Assemblies

AU - Insinga, Andrea Roberto

PY - 2016

Y1 - 2016

N2 - The aim of this thesis is to investigate a framework to design and optimise magnetostatic systems. Over the course of the last decades the range of applications of permanent magnets expanded considerably, thanks to the development of powerful rare-earth permanent magnets. Concurrently, the research on methods to optimise permanent magnet based magnetic systemsintensified. The increase in computational power, and the emergence of new optimisation algorithms provided new instruments for the design of magnetic systems. All these factor contribute in making the optimisation of magnetic systems a very lively sector of modern research.The main focus of this work are magnetic systems based on permanent magnets, although hybrid systems combining permanent magnets with electromagnets are also considered. Many optimisation approaches presented here are derived within a framework based on the reciprocity theorem. This theorem formulates an energy equivalence principle with several implications concerning the optimisation of objective functionals that are linear with respect to the magnetic field. Linear functionals represent different optimisation goals, e.g. maximising a certain component of the field averaged over a region of space. In general, a linear functional can be expressed as the integral over a given region of the scalar product between the magnetic field and an arbitrarily defined objective vector field. It has been known for some time that the reciprocity theorem can be used to determine the optimal remanence distribution with respect to a linear objective functional.Additionally, it is shown here that the same formalism can be applied to the optimisation of the geometry of magnetic systems. Specifically, the border separating the permanent magnet from regions occupied by air or soft magnetic material can be optimised within this framework. Since in the practice most structures are realized by assembling uniformly magnetized pieces of permanent magnet, it is relevant to address the question of how a given region of space is best subdivided. This problem is investigated here within the framework of the reciprocity theorem. Analytical derivations will be used to show that, for segmentations controlled by a single parameter, the globally optimal solution to this problem can be determined for almost arbitrary geometries. The case of segmentations depending by two parameters has been approached employing a heuristic algorithm, which led to new design concepts. Some of the procedures developed for linear objective functionals have been extended to non-linear objectives, by employing iterative techniques.Even though most the optimality results discussed in this work have been derived analytically, the different approaches have been implemented in combination with finite element methods, resulting in flexible and computationally efficient algorithms. Most of the optimisation approaches could only be proven under the assumption of linear magnetic behavior. The last part of this thesis also investigates some of the effects on the performance of magnetic systems, due to non-linear magnetic phenomena. In particular, the non-linear demagnetization effects caused by the finite coercivity of the permanent magnet material will be examined.All the optmisation techniques will be illustrated with example magnetic systems for different applications, thus showing the versatility and efficacy of the various approaches. The Halbach cylinder geometry, relevant for many applications, will be often used as example, also because of the many symmetries and optimality properties exhibited by this geometry. Despite the fact that this system has already been subject of many publications, some of the aspects considered in this thesis have not been investigated before. The ultimate goal of the PhD project is to apply the optimisation techniques developed during this research to the design of the magnetic system for the prototype of heat pump based on the magnetocaloric effect. Magnetic systems for room temperature magnetic refrigeration will thus frequently be used as illustrative examples along the course of this thesis.Primarily because of the theoretical relevance of linear functionals, the results presented here lead to a deeper understanding of the magnet optimisation process. One of the perspectives considered in this work is the trade-off between field intensity and field quality, as the choice of a particular optimisation approach may favour one or the other. The general framework discussed here provides a set of useful tools aiding the magnet design process. This research also opened new scientific questions which would be worth investigating in future studies.

AB - The aim of this thesis is to investigate a framework to design and optimise magnetostatic systems. Over the course of the last decades the range of applications of permanent magnets expanded considerably, thanks to the development of powerful rare-earth permanent magnets. Concurrently, the research on methods to optimise permanent magnet based magnetic systemsintensified. The increase in computational power, and the emergence of new optimisation algorithms provided new instruments for the design of magnetic systems. All these factor contribute in making the optimisation of magnetic systems a very lively sector of modern research.The main focus of this work are magnetic systems based on permanent magnets, although hybrid systems combining permanent magnets with electromagnets are also considered. Many optimisation approaches presented here are derived within a framework based on the reciprocity theorem. This theorem formulates an energy equivalence principle with several implications concerning the optimisation of objective functionals that are linear with respect to the magnetic field. Linear functionals represent different optimisation goals, e.g. maximising a certain component of the field averaged over a region of space. In general, a linear functional can be expressed as the integral over a given region of the scalar product between the magnetic field and an arbitrarily defined objective vector field. It has been known for some time that the reciprocity theorem can be used to determine the optimal remanence distribution with respect to a linear objective functional.Additionally, it is shown here that the same formalism can be applied to the optimisation of the geometry of magnetic systems. Specifically, the border separating the permanent magnet from regions occupied by air or soft magnetic material can be optimised within this framework. Since in the practice most structures are realized by assembling uniformly magnetized pieces of permanent magnet, it is relevant to address the question of how a given region of space is best subdivided. This problem is investigated here within the framework of the reciprocity theorem. Analytical derivations will be used to show that, for segmentations controlled by a single parameter, the globally optimal solution to this problem can be determined for almost arbitrary geometries. The case of segmentations depending by two parameters has been approached employing a heuristic algorithm, which led to new design concepts. Some of the procedures developed for linear objective functionals have been extended to non-linear objectives, by employing iterative techniques.Even though most the optimality results discussed in this work have been derived analytically, the different approaches have been implemented in combination with finite element methods, resulting in flexible and computationally efficient algorithms. Most of the optimisation approaches could only be proven under the assumption of linear magnetic behavior. The last part of this thesis also investigates some of the effects on the performance of magnetic systems, due to non-linear magnetic phenomena. In particular, the non-linear demagnetization effects caused by the finite coercivity of the permanent magnet material will be examined.All the optmisation techniques will be illustrated with example magnetic systems for different applications, thus showing the versatility and efficacy of the various approaches. The Halbach cylinder geometry, relevant for many applications, will be often used as example, also because of the many symmetries and optimality properties exhibited by this geometry. Despite the fact that this system has already been subject of many publications, some of the aspects considered in this thesis have not been investigated before. The ultimate goal of the PhD project is to apply the optimisation techniques developed during this research to the design of the magnetic system for the prototype of heat pump based on the magnetocaloric effect. Magnetic systems for room temperature magnetic refrigeration will thus frequently be used as illustrative examples along the course of this thesis.Primarily because of the theoretical relevance of linear functionals, the results presented here lead to a deeper understanding of the magnet optimisation process. One of the perspectives considered in this work is the trade-off between field intensity and field quality, as the choice of a particular optimisation approach may favour one or the other. The general framework discussed here provides a set of useful tools aiding the magnet design process. This research also opened new scientific questions which would be worth investigating in future studies.

M3 - Ph.D. thesis

BT - Optimising Magnetostatic Assemblies

PB - Department of Energy Conversion and Storage, Technical University of Denmark

ER -