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Theory and simulation of electromagnetic dampers for earthquake engineering applications Haji Akbari Fini, Siavash 2016

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    THEORY AND SIMULATION OF ELECTROMAGNETIC DAMPERS FOR EARTHQUAKE ENGINEERING APPLICATIONS  by   Siavash Haji Akbari Fini   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)   February 2016   © Siavash Haji Akbari Fini, 2016    ii  Abstract The present study develops applications of electromagnetic devices in Civil Engineering. Three different types of electromagnetic system are investigated through mathematical and numerical models. Chapter 3 deals with Coil-Based Electromagnetic Damper (CBED). CBEDs can operate as passive, semi-active and active systems. They can also be considered as energy harvesting systems. However, results show that CBEDs cannot simultaneously perform as an energy harvesting and vibration control system. In order to assess the maximum capacity of CBEDs, an optimization is conducted. Results show that CBEDs can produce high damping density only when they are considered as a passive vibration control system.  Chapter 4 deals with the development of a novel Eddy Current Damper (ECD). The eddy current damper uses permanent magnets arranged in a circular manner to create a strong magnetic field, where specially shaped conductive plates are placed between the permanent magnets to cut through the magnetic fields. Detailed analytical equations are derived and verified using the finite element analysis program Flux. The verified analytical models are used to optimize the damper design to reach the maximum damping capacity. The analytical simulation shows that the proposed eddy current damper can provide a high damping density up to 2,733 kN-s/m4. The Hybrid Electromagnetic Damper (HEMD) are developed and designed in Chapter 5. The idea is to couple the CBED and ECD with the aim of designing a semi-active, active and energy harvesting electromagnetic damper. The simulation results show that it is feasible to manufacture hybrid electromagnetic dampers for industrial applications.     iii  Preface   The author of this thesis is responsible for the literature review, model development computational analysis, data processing, and result presentation.                       iv  Table of contents  Abstract .................................................................................................................................... ii Preface  ................................................................................................................................... iii Table of contents .................................................................................................................... iv List of tables ........................................................................................................................... vii List of figures ........................................................................................................................ viii Acknowledgments ....................................................................................................................x Dedication ............................................................................................................................... xi 1. Introduction ..........................................................................................................................1 1.1 Introduction ...................................................................................................................1 1.2 Literature review ...........................................................................................................4 1.3 Scope of thesis ...............................................................................................................5 1.4 Overview of thesis .........................................................................................................6 2. Review of electromagnetic theory .......................................................................................7 2.1 Introduction ...................................................................................................................7 2.2 Electromagnetic theories ...............................................................................................7 2.2.1 Maxwell’s equations ..............................................................................................7 2.2.2 Lorentz’s law ..........................................................................................................9 2.2.3 Ohm’s law ..............................................................................................................9 2.2.4 Constitutive equation of permanent magnet ..........................................................9 2.3 Simplified electromagnetic damper .............................................................................11 2.4 Conclusion ...................................................................................................................13 3. Coil-based electromagnetic damper .................................................................................14 3.1 Introduction .................................................................................................................14  v  3.2 Mathematical model for coil-based electromagnetic damper .....................................16 3.2.1 Operation point of permanent magnets ................................................................16 3.2.2 Induced voltage ....................................................................................................19 3.2.3 Damping force ......................................................................................................21 3.2.4 Passive, semi-active and active CBED ................................................................22 3.3 Finite element analysis ................................................................................................25 3.4 Optimization procedure ...............................................................................................26 3.5 Conclusion ...................................................................................................................34 4. Development of eddy current damper ..............................................................................35 4.1 Introduction .................................................................................................................35 4.2 Analytical model of the proposed eddy current damper .............................................37 4.2.1 Magnetic flux density produced by an arc segment permanent magnet ..............37 4.2.2 Eddy current damping force calculation ..............................................................39 4.3 Validation of the analytical model using finite element analysis ................................46 4.4 Optimization ................................................................................................................48 4.5 Thermal analysis ..........................................................................................................50 4.6 Conclusion ...................................................................................................................56 5. Electromagnetic dampers with applications in structural engineering ........................57 5.1 Introduction .................................................................................................................57 5.2 Coil-based electromagnetic damper ............................................................................59 5.3 Eddy current damper ...................................................................................................61 5.4 Hybrid electromagnetic damper ..................................................................................62 5.5 Conclusion ...................................................................................................................65 6. Summary and conclusion ...................................................................................................66 6.1 Conclusion ...................................................................................................................66  vi  6.2 Suggestions for future research ...................................................................................68 Bibliography ...........................................................................................................................69 Appendices ..............................................................................................................................73 Appendix A: Analytical expression for magnetic potential ..................................................73 Appendix B: Details of finite element model .......................................................................74     vii  List of tables  Table 3-1. Material properties. .................................................................................................................................... 19 Table 3-2. Damper parameters for the finite element analysis model. ........................................................................ 25 Table 3-3. Optimum parameters for the CBED. .......................................................................................................... 30 Table 4-1. Device parameters. ..................................................................................................................................... 50 Table 4-2. Optimal design values. ............................................................................................................................... 50 Table 4-3. Flow and thermal conditions. ..................................................................................................................... 55 Table 5-1. Optimum design parameters for the CBED. ............................................................................................... 60 Table 5-2. Comparison between bench mark damper and CBED. .............................................................................. 61 Table 5-3. Optimum parameters for ECD. .................................................................................................................. 61 Table 5-4. Comparison between CBED and ECD. ...................................................................................................... 62 Table 5-5. HEMD parameters. ..................................................................................................................................... 64 Table 5-6. Comparison between developed dampers. ................................................................................................. 65 Table 5-7. A summary of the developed electromagnetic dampers. ............................................................................ 67       viii  List of figures  Figure 1-1. Different typs of electromagetic dampers: (a) CBED, (b) ECD. ................................................................ 3 Figure 2-1. Conductive wire and magnetic field: (a). 3D view, (b). Section view. ....................................................... 8 Figure 2-2. Electrical field induced in a conductive wire. ............................................................................................. 8 Figure 2-3. Lorentz’s law. ............................................................................................................................................. 9 Figure 2-4. The relationship between magnetic field and magnetic flux inside a permanent magnet. ........................ 11 Figure 2-5. Simplified electromagnetic damper. ......................................................................................................... 12 Figure 3-1. Coil-based electromagnetic damper: (a) Entire damper, (b) Stationary part, (c) Mover. .......................... 15 Figure 3-2. The two pole CBED: (a). 3D view, (b). Section view. .............................................................................. 17 Figure 3-3. Faraday’s law for the two pole CBED: (a). Two pole machine, (b). Magnetic flux. ................................ 20 Figure 3-4. Cross section of a coil. .............................................................................................................................. 21 Figure 3-5. CBED: (a). Passive mode, (b). Semi-active mode, (c). Active mode. ...................................................... 23 Figure 3-6. Force-displacement relationship for a typical passive CBED. .................................................................. 24 Figure 3-7. Mathematical and FEM results for the magnetic flux density inside a magnet......................................... 26 Figure 3-8. Mathematical and FEM results for the magnetic flux density inside a coil. ............................................. 26 Figure 3-9. The two pole system: (a). Two pole system with three coils, (b). Equivalent one coil system, (c). Active coil wire. ............................................................................................................................................................ 29 Figure 3-10. Optimization procedure........................................................................................................................... 31 Figure 3-11. Penalty value for genetic algorithm. ....................................................................................................... 31 Figure 3-12. Additional resistance effects on CBED. .................................................................................................. 33 Figure 3-13. A four phase CBED. ............................................................................................................................... 34 Figure 4-1. ECD configurations: (a). Conventional ECD (Ebrahimi et al., 2008), (b). Proposed ECD. ..................... 37 Figure 4-2. An arc segment of the permanent magnet. ................................................................................................ 38 Figure 4-3. Conductive plate subjected to one row of permanent magnets on both sides. .......................................... 39 Figure 4-4. Y- Component of magnetic flux at the mid-plane of a sample conductive plate. ...................................... 41 Figure 4-5. Illustration of the pole projection area when a conductive plate is subjected to a uniform magnetic flux. ........................................................................................................................................................................... 42 Figure 4-6. Eddy current density over the conductive plate with finite length. ........................................................... 44 Figure 4-7. A conductive plate subjected to two magnetic fields: (a). 3D view, (b). Mathematical model. ................ 45 Figure 4-8. Eddy currents in a conductive plate. ......................................................................................................... 46 Figure 4-9. Sample comparison of the magnetic flux in the azimuthal and radial directions calculated using analytical and FEM simulations. ........................................................................................................................ 47 Figure 4-10. Plots of velocity versus time and power loss versus time for the ECD under the dynamic load. ........... 48 Figure 4-11. Convergence of genetic algorithm. ......................................................................................................... 49 Figure 4-12. Forced convection internal Flow, (a). 3D view, (b). Cross section. ........................................................ 51 Figure 4-13. Equivalent hydraulic circular section for infinite channel. ..................................................................... 53  ix  Figure 5-1. Hybrid electromagnetic damper. ............................................................................................................... 59 Figure 5-2. Four phase CBED. .................................................................................................................................... 60     x  Acknowledgments   I would like to express my deep gratitude to my survivors Professor Tony Yang and Professor Kang Li for their valuable support and enthusiastic encouragement. Their understanding and willingness to dedicate their time so generously has been a constant motivation for me to tackle all obstacles and achieve the standards of the academic research. It is my pleasure to thank the members of the Smart Structure Research Group at UBC and my friends and colleagues, who helped me with their comments and resources. My thanks also go to all the wonderful staff of Graduate Office of Civil Engineering Department of UBC. Last but not least, I would like to thank my parents who gave me their true love and support in every step of my way.         xi  Dedication  To my beloved parents for their unconditional love and supports  &  In memorial of my best friend, Sina who I missed his smile forever                          1  1. Introduction  1.1 Introduction  In conventional seismic design, the sudden surge of the earthquake energy is dissipated through yielding of the structural elements. This result a significant damage to the structure which leads to hefty financial losses and prolong repair time. In recent years, considerable attentions have been paid to research and development of structural control systems that can improve the performance of structure under extreme earthquake loads. Vibration control systems can be divided into four groups: 1. Passive  2. Active  3. Semi-active  4. Hybrid Passive control systems are realized by adding energy dissipating devices and/or isolating the structures from earthquake ground motion. Passive systems do not require an external power source. Such systems are widely used because of their simplicity and reliability. However, they are effective only over a limited operating range dictated by the tuning condition of their parameters. An active control system is defined as a system which typically requires a large power source for operation. Control forces are developed based on feedback from sensors that measure the excitation and/or the response of the structure. By using the active vibration control, the dynamic characteristic of a system can be altered, hence a greater reduction in vibration levels can be achieved. However, an incorrectly designed active control system can lead to an increased level of vibration in the controlled system.  2  A semi-active control system requires a small external power source for operation. This system utilizes the motion of the structure to develop the control forces. The magnitude of forces can be adjusted by the external power source. Similar to active control systems, their mechanical properties are typically adjusted based on measured feedback from the structural.  The three major classes of control systems described above can be combined to form hybrid control systems. Hybrid control systems consist of combined passive and active devices or combined passive and semi-active, and have been designed to take advantages of passive, active and semi-active control systems. Electromagnetic damper is known as one of the most recent developed vibration control system. Electromagnetic dampers can operate as passive, semi-active and active systems, or can function in combination with other devices as hybrid systems. In general, there are two types of electromagnetic dampers (see Figure 1-1), Coil-Based Electromagnetic Damper (CBED) and Eddy Current Dampers (ECD). In both systems, electrical current and the damping force are generated when relative motion happens between the permanent magnets and the conductive material. In CBED, the electrical current is diverted into the electrical coil and can be stored in a battery, while in the ECD the electrical current will appear in a swirling pattern in the conductive material. The swirling electrical current is commonly known as the eddy current, hence this type of damper is known as ECD. The following section presents a brief literature review of previous study.   3     Figure 1-1. Different typs of electromagetic dampers: (a) CBED, (b) ECD.     4  1.2 Literature review Applications of electromagnetic devises have been explored over the past decades. Coil-based electromagnetic systems can be employed as both micro scale generators and/or large scale dampers. Lu et al., (2005) developed a tubular electromagnetic micro-motor for robotic applications. Buren & Troster, (2007) designed a micro power generator to supply power for sensors. Ebrahimi et al., (2008) and Palomera-Arias et al., (2008) studied the application of CBEDs as a vibration control systems in vehicles and structures. Nakamura et al., (2014) have proposed a new type of CBED with rotating inertial mass to control the vibrations of structures. Wang & Hua, (2013) studied the application of the passive CBED for bridge stay cables through experimental tests.  The application of CBEDs for energy harvesting has been recently investigated. Zhu et al., (2012) discussed the dynamic characteristic of CBED for both energy harvesting and damping force. In addition, Shen & Zhu, (2015) investigated the CBED for energy harvesting in a bridge stay cables. Zuo et al., (2010) also proposed a new type of electromagnetic energy harvester for vehicle suspension systems. Eddy current systems are employed for different applications. Lee & Park, (1999) applied the eddy current phenomena to propose a non-contact braking system. Bae et al., (2005) and Sodano et al., (2005) studied the application of an ECD in controlling the vibrations of small cantilever beam. Kienholz et al., (1996) used the eddy current systems as an isolation system for a space shuttle payload. Ebrahimi et al., (2008) proposed an ECD for the vehicle suspension systems. Wang et al., (2012) investigated the feasibility of using eddy current devices in tuned mass dampers (TMD).   5  Efforts have been made to improve the damping performance of ECDs. For instance, Zuo et al., (2011) and Ebrahimi et al., (2010) studied the influence of permanent magnets arrangements to increase the damping density of ECDs. Sodano et al., (2006) studied the influence of different type of magnetic flux cycle to improve the performance of ECDs. Hybrid electromagnetic damper (HEMD) is also investigated to take the advantages of both CBED and ECD. Gysen et al., (2010) utilized the coli spring and the electromanetic system to improve the vehicle suspension systems. Martins et al., (1999) studied the hybrid devices integrating the active and passive systems.  1.3 Scope of thesis  In Chapter 3 the performance of CBEDs is analyzed by means of detailed mathematical and finite element method (FEM). A novel ECD with circumferential permanent magnets is proposed in Chapter 4. Detailed analytical model of the ECD is derived using electromagnetic theory under quasi-static condition. The dynamic response of ECD is also verified using FEM. In addition, a genetic algorithm (GA) is applied to optimize the CBED and ECD capacity. A HEMD is proposed in Chapter 5 to integrate the features of CBED and ECD. The contributions of the present study are summarized below: 1. Performance evaluation of CBEDs 2. Investigation on the optimum configuration and maximum capacity of CBEDs 3. Development of a novel ECD with high damping density 4. Quantifying the magnetic field produced by an arc segment permanent magnet 5. Investigation on the optimum configuration and maximum capacity of ECDs 6. Proposing the HEMD   6  1.4 Overview of thesis The material presented in the remainder of this thesis is organized into five chapters. Chapter 2 briefly reviews the electromagnetic theories. In Chapter 3, the CBEDs are studied in details through mathematical and numerical models. In addition, the maximum capacity and applications of CBEDs are discussed. In Chapter 4, a new type of ECD is developed. As will be shown, the proposed ECD can provide high damping force comparable to a viscous damper. In Chapter 5, the possible application of electromagnetic systems are evaluated and compared with the benchmark damper. The HEMD is also proposed in Chapter 5 to integrate feature of CBED and ECD. Finally, Chapter 6 summarizes the findings, and presents suggestions for future studies.              7  2. Review of electromagnetic theory 2.1 Introduction This chapter presents electromagnetic theories for the quasi-static magnetic field condition. The quasi-static condition describes a slowly varying magnetic field. This condition can be considered for the design of electromechanical devices which operate at low frequencies and relatively low velocities. For a better understanding of electromagnetic theories, the mathematical model for a simplified electromagnetic damper is also derived here.  2.2 Electromagnetic theories  2.2.1 Maxwell’s equations  Equations (2-1) to (2-3) define the Maxwell’s equations under the quasi-static magnetic field condition (Furlani, 2001): . .C SH dl J ds      Ampere’s law    (2-1) . .C SBE dl dst          Faraday’s law     (2-2) . 0SB ds      Continuity condition    (2-3) Where H , J , E  and B  define the magnetic field strength, current density, electrical field and magnetic flux density, respectively. t is a partial derivation with respect to time.  Ampere’s law explains the relationship between the magnetic field strength in space and the electrical current passing through a conductor. Figure 2-1 shows a conductive wire and the magnetic field produced by a wire. According to Ampere’s law, the amount of a magnetic field at an arbitrary line, for example contour C in Figure 2-1, is determined by integrating the current density over the surface which is defined by contour C (equation 2-1).  8   Figure 2-1. Conductive wire and magnetic field: (a). 3D view, (b). Section view.  Faraday’s law determines the induced electrical field due to the time-varying magnetic field. When the amount of magnetic flux varies in a closed loop, for instance contour C in Figure 2-2, the electrical field will be generated in the conductor. The electrical field is defined by the integral over the contour C and the surface defined by C (equation 2-2).   Figure 2-2. Electrical field induced in a conductive wire.   9  2.2.2 Lorentz’s law Lorentz’s law quantifies the amount of damping force. When a particle electrical charge, q, with the velocity v  moves in the magnetic field, the particle will be subjected to the force. The force is equal to cross product of velocity and magnetic field as: ( )F q v B           (2-4) Where F  is known as the electromagnetic force. Figure 2-3 shows a conductive wire in a magnetic field. Integrating the cross product of current density and magnetic field over the wire volume, V, defines the electromagnetic force as:  ( )VF J B dV     Lorentz’s Law     (2-5)  Figure 2-3. Lorentz’s law.  2.2.3 Ohm’s law The Ohm’s law describes the relationship between the current density and electrical field in a conductive material as:  J E           (2-6) Where   defines the electrical conductivity of material.   2.2.4 Constitutive equation of permanent magnet Following Haus et al., (2008), the constitutive equation for magnetic flux density inside a permanent magnet, Bm, is: 0 ( ) m mB H M          (2-7)  10  Where 0  is the permeability of the free space, Hm is the magnetic field inside the permanent magnet and the M is the magnetization of the permanent magnet. Magnetization is a function of magnetic field, and whenever the magnetic field is removed, Hm=0, it will be equal to the initial magnetization as M0. Figure 2-4 shows the relationship between Bm and Hm. In other words, when permanent magnets are located in free space or in the domain with non-magnet material such as copper, the magnetic flux density inside the magnets is only proportional to M0 as: 0 0rB M           (2-8) Where Br is the residual magnetic flux density defined in Figure 2-4. However, when permanent magnets operate in the presence of ferromagnetic material such as iron, the magnetic flux, Bm, cannot be defined by equation (2-7). In this condition, the magnets operate as a source of energy and magnetize the ferromagnetic material. As a consequence, the magnets are demagnetized and operate in the second quarter of Figure 2-4. It is more convenient to approximate the demagnetization curve using the straight line as: m r m mB B H           (2-9) rmcBH            (2-10) Where Br and Hc define the residual magnetic flux density and coercivity magnetic field as shown in Figure 2-4.  The above-mentioned equation can be generalized to define the relationship between the magnetic field and magnetic flux density inside all materials using equations (2-11) and (2-12): B H           (2-11) o r             (2-12)  11  Where   and r  are the permeability and relative permeability of material. r  for the non-magnet material is approximately equal to 1 and for the ferromagnetic material is much greater than unity.   Figure 2-4. The relationship between magnetic field and magnetic flux inside a permanent magnet.   2.3 Simplified electromagnetic damper The electromagnetic theories are applied to derive a mathematical expression for a simplified electromagnetic damper. Figure 2-5 illustrates a simplified electromagnetic damper. The device consists of a conductive loop with the length and width of x and L, respectively. It is also assumed that the cross sectional area of the mover part is 1 m2. The system is subjected to a uniform magnetic flux density, B, as illustrated in Figure 2-5.The translation motion of the mover causes a variation in the magnetic flux inside the loop. Consequently, the current is induced in the conductors. According to Lorentz’s law, when the conductor carrying a current is located in a  12  magnetic field, it will be subjected to a force. The governing equations will be described in the following paragraphs.   Figure 2-5. Simplified electromagnetic damper.  The magnetic flux, TB passes the conductive loop is given by equation (2-13).  .BT B dA            (2-13) Where B defines the magnetic flux density over the loop as shown in Figure 2-5. Considering Figure 2-5, the total magnetic flux is equal to equation (2-14). BT BLx            (2-14) According to Faraday’s law (equation 2-2), the variation of magnetic flux with respect to the time produces an electrical field as stated in equation (2-15).  BTEt          (2-15)  13  Substituting equation (2-14) into equation (2-15) the electrical force can be defined by equation (2-16).    E BLx BLvt        (2-16) Where v is the velocity. The current density, J, is defined using the Ohm’s law (equation 2-6) and equation (2-16) as:   E BLvJR R         (2-17) Where R (1/ ) represents the resistance of the conductive loop. The amount of damping force is quantified using Lorentz’s law (equation 2-5) as: ( ) VF J B dV          (2-18) Where V is the mover volume. Substituting equation (2-17) into (2-18) the damping force for the simplified electromagnetic damper can be rewritten as: 2( )BLF vR          (2-19) The damping coefficient, C, in this case clearly appears as: RBLC2)(           (2-20) 2.4 Conclusion In this chapter, the fundamental electromagnetic theories are described under the quasi-static condition. In addition, a mathematical expression is derived for a simplified electromagnetic damper. In the following chapters the electromagnetic theories will be applied to investigate the CBED and ECD.    14  3. Coil-based electromagnetic damper 3.1 Introduction The loss of life and infrastructure damage from earthquakes around the world reveals the importance of understanding and controlling the structural responses. In the past decades, significant efforts have been devoted to developing new vibration control systems such as viscous and visco-elastic dampers (Samali & Kwok, 1995), buckling restrained brace systems (Yang et al., 2014) and MR dampers (Yang et al., 2002). In all common dampers, the vibration energy is dissipated by converting into heat. Coil-Based Electromagnetic Damper (CBED) is considered to be one of the most recent developed vibration control systems. CBEDs can be utilized as a damper or energy harvester system. The CBED converts the kinetic energy into electricity that can be stored in an external battery through an energy harvesting circuit attached to a damper. This chapter investigates the performance and optimum configuration of CBEDs. Figure 3-1 illustrates the machine configuration. In this configuration, the mover consists of axial permanent magnets while coils are stationary and located in the damper shell. The damping force is produced as a result of the relative movements of permanent magnets and coils. Furthermore, considering Faraday’s law (equation 2-2), the voltage is induced in the coils. This chapter is divided into two parts. The first part derives the mathematical model for EMDs. In general, three approaches are available to express the mathematical model:  1. Lumped equivalent magnetic circuit applied by Lu et al., (2005), Ebrahimi et al., (2008) and Palomera-Arias et al., (2008). 2. Numerical solution applied by Basak & Shirkoohi, (1990) and Zuo et al., (2010). 3. Analytical solution, separation of variables, which applied by Wang et al., (2004) and Tsai & Chiang, (2010).  15  The first method is utilized here to derive the mathematical model. The formulation is then verified by finite element analysis. In the second part, the genetic algorithm in Matlab toolbox (Mathworks, 2013) is used to investigate the maximum capacity of CBEDs using a verified mathematical model. Results show that the maximum achievable damping density for the CBED is 3,061 kN-s/m4. In addition, corresponding damper details for the optimum design configuration are provided.   Figure 3-1. Coil-based electromagnetic damper: (a) Entire damper, (b) Stationary part, (c) Mover.   16  3.2 Mathematical model for coil-based electromagnetic damper 3.2.1 Operation point of permanent magnets The permanent magnets are used in the design of CBEDs. A magnetic circuit will be analyzed to calculate the damping force and induced voltage. Magnetic circuit is similar to electrical circuit, however in a magnetic circuit a permanent is used as a source of energy instead of a battery in an electrical circuit. To analyze a magnet circuit, the operation point of permanent magnets should be determined. The operation point defines the amount of energy produced by permanent magnets in a circuit.  Figure 3-2 shows a two pole CBED with two active coils. An active coil is a coil which is subjected to the magnetic flux. The magnet circuit analysis is conducted for a two pole machine to determine the operation point of permanent magnets. It should be noted that the results can be utilized for a several pole device. Ampere’s law (equation 2-1) is applied over the magnetic flux closed loop as indicated in Figure 3-2(b). Since the active coils experience different magnetic fields both upward and downward, the direction of the currents inside the active coils run in opposite directions, which means the left side of Ampere’s law is zero.  17    Figure 3-2. The two pole CBED: (a). 3D view, (b). Section view.  And the right side of Ampere’s law is: . 2 2 2m pole air coil shellC Magnet Pole Air Coil ShellH dl H dl H dl H dl H dl H dl            (3-1) Where H and its subscript describes the magnetic field in various regions. It should be noted that the magnetic field is axial inside the magnet and shell and is radial inside the iron pole, air gap and coils. Using the consecutive law of material (equations 2-11 and 2-12) above equation can be modified as: 0 0 0 00. 2 2 2 ( )m a cm ar r rpole air coil shellm m m iFe cu FeC r rB B B BH dl H t dr dr dr t t                 (3-2)  18  Where B and its subscript describes the magnetic field in various regions. cu  and Fe  are the relative permeability of copper and iron. Table 3-1 shows the material properties used in this study. Also rm, ra, rc, tm and ti are the dimensions described in Figure 3-2(b). Using equation (2-9), Hm can be defined in terms of Bm:  m rmmB BH          (3-3) The continuity condition, equation (2-3), can be applied to determine the magnetic flux density in the different regions in terms of Bm:        02mm m pole pole pole mirBA B A B B r rt          (3-4) 2       2m mm m air air air m air BA B A B B r r rrt          (3-5) 2       2m mm m coil coil coil a cir BA B A B B r r rrt          (3-6) 22 2 ( )m mm m shell shell shells cr BA B A B Br r        (3-7) The integrals in equation (3-2) can be calculated as: 20 004mrpole mm pole mFe Fe iB rdr B k Bt           (3-8) 20 0ln2amrair m am air mi mrB r rdr B k Bt r            (3-9) 20 0ln2carcoil m cm coil mcu cu i arB r rdr B k Bt r              (3-10) 22 20 0( )( )shell mm i m shell mFe s c FeB rt t B k Br r          (3-11)  19  Substituting equations (3-8) to (3-11) in equation (3-2) the operation point of permanent magnets is derived as: (2 2 2 )r mmm m pole air coil shellB tBt k k k k         (3-12) The derived Bm defines the amount of magnetic flux produces by a permanent magnet in a CBED. Table 3-1. Material properties. Parameters Description Values 0  Permeability of Free Space 74 10   cu  Relative Permeability of Copper  1  Fe  Relative Permeability of Iron 2000  rB  Residual Magnetic Flux Density 1.2  cH  Coercivity Magnetic Field 3900 10    Electrical Conductivity  75.65 10   3.2.2 Induced voltage Faraday’s law (equation 2-2) is applied to determine the induced voltage in the CBED shown in Figure 3-3(a). Equation (3-13) shows the left side of Faraday’s law for a single arbitrary wire at the (x, r) position in the active coil region shown in Figure 3-3(b).   B . 2 2               coil coil coil a cdds B rx B rv r r rt dt     (3-13) Where Bcoil is defined in terms of Bm using equation (3-6) and v is the velocity. The right side of Faraday’s law describes the induced voltage. The induced voltage for a single wire, Ewire, can be stated by substituting equation (3-6) in equation (3-13) as:   20  2 m mwireir BE vt         (3-14) As a result, for the Np active coli and Na active wire turns in each coil the total induced voltage, EEM is:  2 m mEM p air BE N N vt        (3-15)  Figure 3-3. Faraday’s law for the two pole CBED: (a). Two pole machine, (b). Magnetic flux.  The resistant of the coil should also be determined to define the current in the electromagnetic damper. The resistance of a wire, R, with a length and cross-sectional area of l and Awire is: wirelRA           (3-16) Where   is the electrical conductivity. Considering Figure 3-4 the average radius, rave, and length, lave, of a single wire used in a coil are: 2a caver rr          (3-17) 2 ( )2a caver rl           (3-18)  21  And the cross-sectional area of a wire is: 2wire wA r          (3-19) Where rw is the radius of a wire. The average resistance can be calculated using equations (3-16) to (3-19) as: 2( )a cavewr rRr          (3-20)   Figure 3-4. Cross section of a coil.  And the total resistance of the device is: 2( )a cEM w cwr rR N Nr         (3-21) Where Nw and Nc define the total number of wire turns in each coil and the number of coils in a CBED, respectively. 3.2.3 Damping force Lorentz’s law (equation 2-5) is applied to quantify the force experienced by the mover part due to electrical currents inside the coils. Equation (2-5) can be rewritten for the force applied to a single wire, Fwire as: ( ) wire coillF JAB dl         (3-22)  22  Where l and A are the length and cross-sectional area of a single wire, and Bcoil is the magnetic flux density inside the coil. JA also is equal to the current, i, inside the coil. Therefore, the force applied to a single wire is: 2 wire coil coillF iB dl riB        (3-23) Bcoil can be explained in terms of Bm using equation (3-6) as:  2       2m mcoil a cir BB r r rrt           (3-24) Substituting equation (3-24) into equation (3-23) the Fwire is: 2 m mwireir B iFt         (3-25) And the total damping force, FEM, is: 2 m mEM p air BF N N it        (3-26) 3.2.4 Passive, semi-active and active CBED The CBED can perform as a passive, semi-active and active vibration control system based on an electrical circuit attached to the damper. In this section, the different modes of damper operation are briefly investigated. The constitutive equations for the damper are repeated here as:  2 m mEM p a EMir BE N N v K vt       (3-27) 2( )a cEM w cwr rR N Nr         (3-28) 2m mEM p a EMir BF N N i K it         (3-29) Where KEM is known as the damper constant.   23  Figure 3-5(a) shows a passive CBED. When the damper operates in a passive mode, the machine parameters cannot be adjusted during the vibration and the circuit equation follows simple Ohm’s law as: EM EME R i          (3-30) It should be noted that the inductance of the coil, LEM, is neglected for the low frequencies vibration such as earthquake excitations (1-10 Hz) and it is not considered in the circuit equation. The damping force for the passive modes of vibration, FPEM, is derived by substituting equations (3-27) and (3-28) in equation (3-29) as: 2EMPEMEMKF vR          (3-31)  Figure 3-5. CBED: (a). Passive mode, (b). Semi-active mode, (c). Active mode.  The CBED can be considered as a semi-active system by adding an external resistance as shown Figure 3-5(b). The circuit equation and damping force are derived using Kirchhoff voltage law as:   EM EM addE R R i         (3-32) And the semi-active damping force, FSEM, is derived using equation (3-27) to (3-28) as:   24   2EMSEMEM addKF vR R        (3-33) Similar to semi-active mode, the CBED can operate as an active system by adding an additional source of energy, ESE, as illustrated in Figure 3-5(c). Like passive and semi-active modes, the damping force in the active mode, FAEM, is:  EM EM SEE R i E          (3-34) 2  EM SEEMAEMEM EMK EKF vR R        (3-35) When the CBED performs as a passive device, its operation can be modeled as a linear damper with a constant damping coefficient (equation 3-31). Figure 3-6 illustrates the force-displacement relationship for a sample passive CBED under harmonic excitation.  Figure 3-6. Force-displacement relationship for a typical passive CBED.    25  3.3 Finite element analysis  The mathematical model for the CBED has been defined in the previous sections. The analytical model for Bm has been stated in equation (3-12). In the present section, a finite element analysis is conducted to validate the previous formulations. A 2D axisymmetric model of the mover is analyzed under the magneto-static condition using finite element software, Flux (Cedrat, 2013). Table 3-2 defines the finite element analysis parameters. Figure 3-7 compares the numerical and mathematical results for the magnetic flux density inside the permanent magnets for the various range of mover to magnet length ratios, RL. Also, when RL is equal to 2, the variation of magnetic flux density inside the coil, Bcoil,, is studied in Figure 3-8. As illustrated, the error between the two models is constant and equal to 2.77 percent.  Table 3-2. Damper parameters for the finite element analysis model. Parameters Description Values rs Damper Shell Radius 121.1 mm rc Coil Radius 109.4 mm ra Air Gap Radius 101 mm rm Magnet Radius 100 mm tl Mover Length 80.5 mm RL Mover to Magnet Length Ratio [1.2-2.5] tm Magnet Length tl/RL ti Iron Pole Length (tl-tm)/2 tc Coil Length ti   26   Figure 3-7. Mathematical and FEM results for the magnetic flux density inside a magnet.   Figure 3-8. Mathematical and FEM results for the magnetic flux density inside a coil.   3.4 Optimization procedure The derived analytical equations quantify the damping coefficient on the basis of mechanical and material properties. The verified formulations are used to investigate the maximum damping capacity of CBED.   27  Since the two pole system is repeated along the CBED (see Figure 3-1), only the two pole damper is considered for the optimization. The measure that is assessed is a damping density, defined as a variable determining the achievable damping force in a unite machine volume. The volume, VEM, and damping coefficient, CPEM, for the passive CBED can be stated using equation (3-36) to (3-37) as: 2222( )( ) m mp aiEMPEMa cEMw cwr BN NtKCr rRN Nr       (3-36) 22pEM l sNV t r          (3-37) The term NwNc defines the total number of coil wire turns in the device which is equal to NpNt/2. Nt defines the number of coil turns between two pole. Figure 3-9(a) and (b) shows the definition of Nc, Nw and Nt. Moreover, the term Na/ti is equal to Nt/tl, since the coil turns are uniformly distributed over the machine length (see Figure 3-9(b) and (c)). Therefore equation (3-36) can be rewritten as: 2222( )( )2 m mp tlEMPEMa cEMwr BN NtKCr rRr       (3-38) Using equations (3-36) and (3-37), the damping density for the passive CBED, CVPEM, is:  4 2322( )4t m mlPEMPEMa cEMswN r BtCCVr rVrr        (3-39) The Nt can be determined by equation (3-40) proposed by Zhu et al., (2012): 2 3ctwANA          (3-40)  28  Where Ac and Aw are the cross-sectional areas of the coil and conductive wire. The genetic algorithm (GA) is Matlab toolbox (Mathworks, 2013) used to optimize the damping density for the range of parameters defined in Table 3-3 and Figure 3-9(a). To ensure accurate results, the population size, generation and tolerance function for GA optimization are considered as 150, 200 and 10E-11, respectively. It should be pointed out that iterations are needed to calculate the damping density in each step of GA. In other words, all the machine parameters can be determined based on the information provided in Table 3-3, except the damper shell radius, rs. Another constraint, magnetic flux saturation condition, is applied to find the minimum required damper shell radius. A magnetic flux saturation condition defines the maximum magnetic flux that can pass through the material in a magnetic circuit. Using equation (3-7), the iron shell radius can be determined as: 22 2 ( )m mm m shell shell shells cr BA B A B Br r        (3-41) Whenever the Bshell is equal to the saturated magnetic flux, Bsat, the minimum damper shell will be reached. Figure 3-10 describes the optimization procedure.    29   Figure 3-9. The two pole system: (a). Two pole system with three coils, (b). Equivalent one coil system, (c). Active coil wire.  30  Table 3-3. Optimum parameters for the CBED. Parameters Description Values Optimum Value Variable    tl Mover Length [10-500] mm 80.5 mm RL Mover to Magnet Length Ratio [1.2-2.5] 1.96 tm Magnet Length tl/RL 41 mm ti Iron Pole Length (tl-tm)/2 19.7 mm tc Coil Length tl 80.5 mm rm Magnet Radius [10-100] mm 100 mm hg Air Gap Thickness [1-5] mm 1 mm ra Air Gap Radius rm+hg 101 mm hw Coil Thickness [10-100] 8.4 mm rc Coil Radius ra+hw 109.4 rs Damper Shell Radius Saturation Condition 121.1 mm rw Wire Radius [0-40] AWG 21 AWG Target Function    CVPEM Passive Damping Density  3,061 kN-s/m4   31   Figure 3-10. Optimization procedure.  Figure 3-11. Penalty value for genetic algorithm.  32  Figure 3-11 and Table 3-3 illustrate the GA iteration and the optimum parameters. The maximum achievable damping density for the passive CBED is 3,061 kN-s/m4 which is in the order of regular oil dampers that are currently used in the structures (2,800-4,200 kN-s/m4 (Taylor, 2003). However, when the CBEDs are considered as semi-active or active systems, the additional resistance or power which is applied to the damper circuit alters the damping force. For instance, in the semi-active control mode the damping force, FSEM, is:  2EMSEMEM addKF vR R        (3-42) Assuming the additional resistance as a portion of damper resistance, the damping force is stated as: add EMR R          (3-43)  21EMSEMEMKF vR         (3-44) Figure 3-12 shows the ratio of semi-active to passive mode damping force. It can be concluded that by adding the additional resistance, which is inevitable during a control algorithm, the damping force is rapidly decreased. A similar discussion is applicable when the CBEDs are employed as an energy harvester. Shen & Zhu, (2015) proposed the optimum additional resistance, ROpt,E to reach to the maximum harvesting capability as: 2, 1EMOpt E EMp EMKR RC R          (3-45) Where Cp is a constant and defines the mechanical losses in CBEDs due to some imperfection such as friction.  33  In light of the above discussions, when the CBEDs operate either as semi-active or active dampers or as a harvesting energy system, the total resistance is no longer equal to REM. Therefore, the damping force significantly affected by additional resistance and will be decreased.    Figure 3-12. Additional resistance effects on CBED.  In the optimization procedure, equation (3-39), a single equivalent coil is used between two poles. However, in practice, when the coil length is equal to the magnetic flux cycles, the voltage will not be generated since both positive and negative voltage will be induced in the same coil segments. As discussed in Table 3-3 the maximum damping density occurs when the mover to magnet length ratio, RL, is approximately 2. It is therefore logical to consider coil length the same as magnet length. As a result, all the coils in the passive CBED perform as a four phase electrical machine. As Figure 3-13 shows, the first and third phases generate the minimum positive and negative voltage at the indicated position, and the second and fourth phases have zero voltage. It  34  is worth to mention that, although the voltage or power of each coil depends on the relative position of coil and mover, the total generated power is constant.  Figure 3-13. A four phase CBED.  3.5 Conclusion  In this chapter, the mathematical model for the CBED is derived and verified by means of finite element analysis. Different applications of CBED as passive, semi-active and active dampers are briefly studied. The verified formulation is optimized through the genetic algorithm (GA) to find the maximum damping density and its corresponding machine configuration. As a result, the four phase CBED is designed which can generate 3,061 kN-s/m4 damping density in its passive mode. As discussed, the capacity of passive CBED is within the range of conventional oil dampers. However, when they are implemented as semi-active and active systems or as an energy harvester, their damping force are no longer enough to be used in structural systems. In chapter 4, another type of passive electromagnetic damper, the Eddy Current Damper (ECD) will be examined. Chapter 4 will suggest ways of improving the performance of electromagnetic dampers for large scale applications such as structural systems.     35  4. Development of eddy current damper  4.1 Introduction In the previous chapter, the concept of Coil-Based Electromagnetic Damper (CBED) was discussed. An optimum device configuration was suggested in order to reach the maximum capacity of CBED. The present chapter explores the development of new types of a passive electromagnetic damper identified as the Eddy Current Damper (ECD). The eddy current phenomena appears due to the relative motion of magnets and conductors which induces an electrical field and therefore a current in the conductors. Considering Lorentz’s law (equation 2-5) the eddy currents produce the reverse force which is treated as a damping force. In other words, the ECD converts the kinetic energy into the heat through the eddy currents in the conductor without any contacts.  Traditional ECD has two significant disadvantages: 1) the ECD usually have small damping density in comparison with other structural dampers (Warmerdam, 2000). 2) The conventional ECD usually requires iron poles and shells to guide the magnetic flux in a loop (see Figure 4-1(a)). The iron can be easily magnetized. During vibration, the iron shell in the conventional ECD is subjected to alternating magnetic fields, which cause the direction of the magnetism within the iron pole to change frequently, and eventually losses its ability to guide the magnetic flux.  In this chapter, a novel ECD design with the circumferential permanent magnets, (see Figure 4-1(b)), is proposed. In the proposed ECD, the permanent magnets are placed in a circumferential pattern, which can produce high magnetic flux without the use of iron, which will produce a higher damping density. Simulations show that the proposed ECD design can produce a high damping density (2,733 kN-s/m4) in the order of viscous fluid dampers (2,400 kN-s/m4 to 4,200 kN-s/m4) (Taylor, 2003).  36  This chapter is divided into three sections. In the first section, the analytical model of the proposed ECD is derived using electromagnetic theory under quasi-static condition. The mathematical model is verified through the finite element method (FEM). In section two, the verified mathematical model is used for the optimization procedure using the genetic algorithm (GA). As previously stated, in ECDs the kinetic energy is converted into heat. A thermal analysis is therefore carried out in section three in order to investigate the permanent magnets temperature.  37    Figure 4-1. ECD configurations: (a). Conventional ECD (Ebrahimi et al., 2008), (b). Proposed ECD.  4.2 Analytical model of the proposed eddy current damper 4.2.1 Magnetic flux density produced by an arc segment permanent magnet  Figure 4-2 illustrates an arc permanent magnet and the coordinate system used in this study. In this arc permanent magnet, the magnetization is formulated in a circumference manner as indicated  38  by the vector M . Equation (4-1) defines the magnetization, M , of the arc permanent magnet in the θ  direction. M θ M           (4-1) Where M is magnitude of the magnetization determined from the property of the permanent magnet. In this study, NdFeB N35 magnet with the magnetization of 9.5E5 Ampere/meter is used in ECD design.   Figure 4-2. An arc segment of the permanent magnet.  Equation (4-2) shows the magnetic potential function,  , of the arc segment of the magnet as illustrated in Figure 4-2.  1 .M 1 M.n( , , )4 4   SVr z dV dSd d        (4-2) Where   is the Nabla operator; d is the relative distance between points m and p (see Figure 4-2); p is a point of interest; m is an arbitrary point in the magnet and n i  is the unit normal vector to  39  the ith surface of the magnet. In Figure 4-2, 1n  and 2n  are the normal vectors to the north and south poles, respectively. Due to the special magnetization pattern shown in Figure 4-2, .M = 0 , 1M.n  M , 2M.n  M  and M.n 0i  for the other surfaces. Using the parameters defined in Figure 4-2, the magnetic potential for the arc magnet is presented in the Appendix A. The associate magnetic field, H , and flux density, B , can be derived using equations (4-3) and (4-4), respectively.  H ( , , )  ( , , ) r z r z           (4-3) 0B( , , ) H( , , )r z r z         (4-4) Where 0  is the permeability of air (70 4 10   ).  4.2.2 Eddy current damping force calculation Figure 4-3 shows a conductive plate placed between two arc permanent magnets. In addition, Figure 4-3 illustrates the dimension and coordinate system used for the following derivations.   Figure 4-3. Conductive plate subjected to one row of permanent magnets on both sides.  40  The current density, J , induced in the conductive plate due to the relative movement of conductive plates and permanent magnets is described in equations (4-5)–(4-7). J= E            (4-5) c bE=(E +E )          (4-6) bE =v×B           (4-7) Where   is the electrical conductivity of the plate (in this study, the copper plate with electrical conductivity of 5.8E7/Ohm is used); cE  is the electrostatic field generated by the coulomb charges, and bE  is the electromotive field induced by the relative movement of the conductive plate at the relative velocity v . B  is the magnetic flux density with rB , θB  and zB  components which can be defined by equation (4-4). Due to the anti-symmetric arrangement of the permanent magnets on both side of the conductive plate, the radial component of magnetic flux, rB , produced by each magnet cancels each other. On the other hand, the azimuthal component of magnetic flux, θB , adds up. Because the relative movement of the conductive plate and the permanent magnets is in the Z direction, the Z component of the magnetic flux density, zB , does not contribute to bE . To simplify the derivation, a new Cartesians coordinate system with the origin O at the center of the pole projection area (see Figure 4-3) is used. Because the air gap is small, the   angle is approximately zero and the vector yB can be approximated using θB . Similarly, xB is almost zero. The relative velocity between permanent magnets and plates is also in Z direction, vz. Therefore, bE (equation 5-7) can be simplified to:  b bxE = E = .z × .yz yv B         (4-8)  41    Figure 4-4 shows a sample illustration of the amount of magnetic flux density, yB , at the mid-plane of a conductive plate (with the parameters as shown below). As shown in this figure, the proposed magnets configuration can produce a high magnetic flux density in the conductive plate.  Figure 4-4. Y- Component of magnetic flux at the mid-plane of a sample conductive plate.  Figure 4-5 shows the pole projection area and the reference axes on a conductive plate when the plate is subjected to a uniform magnetic flux density. Equation (4-9) shows the electrostatic field generated by the coulomb charges in the x-direction, cxE , (Heald, 1988).  , arctan arctan arctan arctan2                                   z avecxv B z b z b z b z bE x zx a x a x a x a   (4-9) Where Bave is the average magnetic flux density over the pole projection area defined using the dimension 2b and 2a as shown in Figure 4-5.  42   Figure 4-5. Illustration of the pole projection area when a conductive plate is subjected to a uniform magnetic flux.  The current density in the x-direction, xJ , inside and outside of the pole projection can be described using equations (4-10) and (4-11), respectively.  ( ) x cx bxJ E E    Inside the pole projection area  (4-10)  x cxJ E     Outside the pole projection area   (4-11) Where bxE  and cxE  are stated by equations (4-8) and (4-9), respectively. The eddy current force, zF , can be calculated using equation (4-12). zF = .x × .y  x yJ B dV         (4-12) It should be note that the amount of eddy current force, zF , can also be expressed as equation (4-13), which is linearly proportional to the relative velocity, zv .    z zF C v           (4-13) Where 2 11 arctan arctan arctan arctan2                                      yz b z b z b z bC Bx a x a x a x a   43  Equation (4-13) is only valid if the conductive plate has infinite length. On the other hand, the conductive plate of the proposed ECD (shown in Figure 4-3) has a finite length and zero eddy current density at its edges. To consider the edges effect, the image method proposed by Lee & Park (2001) is used. In this method, the current density is first calculated using conductive plate with infinite length, then a mirror image of the current density with respect to the edge of the conductive plate is subtracted. Equation (4-14) shows the modified current density for the proposed ECD where the edge effects are included.  ' 1 2 3  x x x xJ J J J          (4-14) Where 1xJ  is the eddy current for the conductive plate with infinite plate; 2xJ  and 3xJ  are the imaginary currents at the two edges of the conductive plate calculated using equations (4-15) and (4-16), respectively.    2 1 12 , x xJ J r x z          (4-15)  3 1 22 ,  x xJ J r x z         (4-16) Where r1 and r2 are the distances to the edges as illustrated in Figure 4-5.  Figure 4-6 shows the eddy current density over a sample conductive plate. The results confirmed that the eddy currents for the finite plate are zero at the edges.   44   Figure 4-6. Eddy current density over the conductive plate with finite length.   Substituting equation (4-14) into equation (4-12), the damping force for the proposed ECD can be calculated using equation (4-17).  1 2 3    b az p x x x yb aF t J J J B dydx        (4-17) Where tp is the plate thickness as shown in Figure 4-3. It should be note that equation (4-17) does not account for the configuration with multiple rows of magnets. To account for multiple rows of magnets (see Figure 4-7), the net electrical field, 'cxE, for the infinite plate can be written as equation (4-18).  ' 1 2,  cx cx cxE x z E E         (4-18) Where 1cxE  and 2cxE  present the electrostatic field produced by the magnets in the first and second row, respectively. 1cxE  is stated using equation (9) and 2cxE  can be defined using equation (4-19).  2 , arctan arctan arctan arctan2                                       avecxvB z b D z b D z b D z b DE x zx a x a x a x a (4-19) Where D is a distance between the center of pole projection areas as illustrated in Figure 4-7(b).  45    Figure 4-7. A conductive plate subjected to two magnetic fields: (a). 3D view, (b). Mathematical model.  Substituting the net electrical field, 'cxE  (equation 4-18), into cxE in equation (4-10), the damping force for the conductive plate subjected to two rows of arc magnetic fields can be described using equation (4-20).  21 2 31      b az p ix ix ix yi b aF t J J J B dzdx       (4-20) Equation (4-21) shows the generalized damping force on the conductive plate subjected to the multiple rows of arc permanent magnets:  1 2 31      MagnetRows b az p ix ix ix yi b aF t J J J B dzdx      (4-21)  46  4.3 Validation of the analytical model using finite element analysis The analytical model is verified using 3D finite element software, Flux (Cedrat, 2013). The permanent magnet is modelled using the Linear Magnetic material with tetrahedral triangle element with automatic mesh generator. The mesh is manually adjusted locally to consider the skin depth effect in the conductive plate. The skin depth effect refers to a phenomenon that the current density at the conductor surface is greater than that at the core (Hayt & Buck, 2001). Figure 4-8 shows a sample 3D finite element result when a conductive plate is subjected to a cyclic displacement loading. As illustrated, the proposed circumferential permanent magnet configuration can produce high current density on the conductive plate.  Figure 4-8. Eddy currents in a conductive plate.   47  Figure 4-9(a) and (b) illustrate the comparison of the analytical model and finite element simulation for the magnetic flux density produced by an arc permanent magnet in the azimuthal and radial directions, respectively. The results show that the error between these two simulations is less than 2.5 percent.   Figure 4-9. Sample comparison of the magnetic flux in the azimuthal and radial directions calculated using analytical and FEM simulations.  Figure 4-10(a-c) show the comparison of the power loss between the analytical and finite element simulation at different loading conditions. The result shows that the analytical model over predicts the power loss compared to the finite element simulation. The discrepancy between these two modeling approaches is highest when the velocity is at its maximum level. Such discrepancy can be attributed by the time variant property of the eddy currents. According to Faraday’s law, because the eddy current is time variant, it will produce additional magnetic fields. It should be noted that this phenomena is not considered in the analytical solution which is based on the quasi-static condition. Therefore, the accuracy of analytical models may be affected by the influence of the time variant property of the eddy currents.   48     Figure 4-10. Plots of velocity versus time and power loss versus time for the ECD under the dynamic load.  4.4 Optimization  Genetic algorithm toolbox as presented in Matlab (Mathworks, 2013) is used to optimize the ECD design to maximize the damping density. The air gap and thickness of conductive plates (as shown in Figure 4-3) are considered as fixed variables due to manufacturing limitations. Table 4-1 shows the range of parameters included in the optimization. In order to carry out an optimization procedure, it is necessary to constrain the device length by defining the number of magnet rows, Nmr. Therefore, a series of optimizations are conducted for the damper with different Nmr.  Figure 4-11 shows a sample convergence of the genetic algorithm. In order to achieve accurate results, the population size, generation and tolerance functions for optimization are considered as 150, 200 and 10E-11, respectively. The result shows the genetic algorithm converged to a solution efficiently. Table 4-2 summarizes the optimal parameters for the range of parameters shown in  49  Table 1 for the different Nmr cases. As illustrated, as Nmr goes above 16, the influence of Nmr values has less influence on the optimal design of the proposed ECD. In addition, the optimum magnet length, tm (see Figure 4-3) converges to 23 mm. The proposed configuration is able to achieve a damping density of 2,733 kN-s/m4. It worth to mention that the damping density for fluid dampers is between 2,400 kN-s/m4 to 4,200 kN-s/m4 (Taylor, 2003). This shows the proposed ECD can have high potential for earthquake engineering applications.   Figure 4-11. Convergence of genetic algorithm.   50  Table 4-1. Device parameters. Parameters Description Values Fixed  gA  Air Gap 1 mm pt  Conductive Thickness 5 mm Variable   θ  (Np) Magnet Angle (Number of Conductive Plate ) [10o-90o] (36-4) 2R  Outer Magnet Radius [10-150] mm 1 2S=R /R  Ratio of Inner to Other Magnet Radius [0-0.95] L  Device Length Defined by Nmr m mrt =L/N  Thickness of Magnets in Z direction [0.5-100] mm  mrN  Number of Magnet Rows [2-100]  Table 4-2. Optimal design values. mrN  θ (Degree)  2R (mm) S  mt (mm) ECCV  (kN-s/m4) 2 30 144.2 0.11 20 2,537 4 32.7 138.4 0.1 21 2,615 8 36 133.6 0.097 21 2,675 12 36 130.2 0.091 23 2,704 16 36 125.5 0.088 23 2,718 25 36 125.5 0.088 23 2,733 50 36 125.5 0.088 23 2,733 100 36 125.5 0.088 23 2,733  4.5 Thermal analysis The eddy current damper converts the kinetic energy into the heat through the induced currents in conductive plates. Due to the high capacity of ECDs, high temperatures are likely to be reached during the operation. All permanent magnet materials are temperature sensitive and it is important  51  to take this into consideration. Heat transfer analysis is therefore required to predict the temperature of the device. Two approaches are available for this purpose: 1. Considering the air flow in different parts of the device by using the computational fluid mechanics applied by Zhao et al., (2015).  2. Simplified analytical models based on empirical equations applied by Negrea & Rosu, (2001).  In the first method, the fluid and electromagnetic system are modeled and solved simultaneously, which is generally time-consuming. In this research the second approach is utilized.  It is assumed that only the air passing through air gaps dissipates the heat in conductive plates, the heat transfer through permanent magnets is not taken into consideration. The device is treated as a steady state forced convection internal flow problem in heat transfer analysis (Incropera et al., 2006). Figure 4-12 describes the assumptions and the cross section of air channels.   Figure 4-12. Forced convection internal Flow, (a). 3D view, (b). Cross section.   52  Newton’s cooling law states the relationships between the heat flux, surface and air temperature as: ( )s fq h T T           (4-21) where q is thermal flux produced by eddy currents in conductive plates, h is a local convection heat transfer coefficient, Ts is the plate surface temperature and Tf  is the ambient temperature which is considered as 293.15o K (25o Celsius).  Whenever the h is defined, it is possible to estimate the plate surface temperature. The convection coefficient is defined using the dimensionless Nusselt number (Incropera et al., 2006). The Nusselt number is well defined for the circular cross section channel for both laminar and turbulence flow conditions. However, many circular channel results can be applied for noncircular channels using the equivalent hydraulic sections. The channel shown in Figure 4-12 is considered as an infinitive plate due to the large length to width ratio of channel cross section. Figure 4-13 shows the equivalent hydraulic section for the infinite channel. The Nusselt number for the equivalent hydraulic section is stated as: hairhDNuk           (4-22) Where kair is the thermal conductivity of the air and is equal to 0.0269 W/m K and Dh is the equivalent hydraulic diameter.  53   Figure 4-13. Equivalent hydraulic circular section for infinite channel.  To quantify the heat transfer coefficient, it is required to define the flow and thermal conditions. The Reynolds number defines the flow condition as:  ReRe<2300  Laminar Flowa hV D        (4-23)  where Va, Dh and   are defined as the air velocity, hydraulic diameter and kinetic viscosity (see Table 4-3).  The thermal condition is considered as a constant surface heat flux condition, since the condition is steady state and the thermal flux due to the eddy currents in the conductive plate is constant. Due to the results in Table 4-3, the flow is treated as a fully developed laminar flow with a constant heat flux. According to the flow and thermal conditions, the Nusselt number for an  54  infinitive channel with one side isolated boundary condition is stated by Bejan, (2013) as 5.39. Therefore, considering equation (4-44) the heat convection coefficient is calculated as 72.5 W/m2 K. The heat flux on the surface of each plate for the maximum damping density, 2,733 kN-s/m4, at the average device velocity, 0.25 m/s, is 2.67E3 W/m2 (see Table 4-3). Using Newton’s cooling law the temperature of plate surfaces is calculated as 62o Celsius, which is within the operation range of permanent magnet, 80o Celsius (MMG, 2006). It should be noted that this temperature is an upper bound estimation, since the radiation and heat transfer through magnets and other parts of the device are neglected.           55  Table 4-3. Flow and thermal conditions. Parameters Descriptions Value Formulation Flow condition    CW Channel Width 1 mm Ag R2 Magnets Outer Radius 68 mm  R1 Magnets Inner Radius 6 mm  CL Channel Length 62 mm R2-R1 Rc Channel Ratio 62 CL/CW Dh Hydraulic Diameter 2 mm 2CW tp Plate Thickness 5 mm  Vp Plate Velocity 0.25 m/s Steady State Condition Va Air Velocity 0.625 m/s Vp.tp/(2CW)- Conservation of Mass   Air Viscosity 1.75E-5 m2/s  Re Reynolds Number Re<2300 VaDh/   Thermal condition    CVEC Maximum Damping Density 2,733 kN-s/m4  qm Maximum Power Density 170k W/m3 CVEC.(Vp)2 A Device Cross Section Area 0.0195 m2  Q Total Power for unit Length 3315 W qm.A Np Number of Plates 10  Ns Number of Plate Surfaces 20 2.Np q Heat Flux for each Plate Surface 2.67E3 W/m2 Q/(Ns.CL)     56  4.6 Conclusion  A novel ECD with circumferential permanent magnets is proposed in this study. The circumferential arrangement of permanent magnets and conductive plates can produce a high strength magnetic field without the use of iron, hence a higher damping force can be achieved. A theoretical model of the proposed ECD is constructed using electromagnetic theory under quasi-static condition. The magnetic flux and eddy current damping force are quantified analytically and validated using finite element simulation. Although the analytical model neglects the effects of the time-variant property of the eddy currents, the analytical model is still able to approximate the damping force of the proposed ECD accurately. The developed analytical model is used to optimize the ECD design using genetic algorithm. The result shows that the proposed ECD can produce a high damping density up to 2,733 kN-s/m4, which is feasible to be used for structural applications. Since the ECDs convert kinetic energy into heat, a thermal analysis for the optimum ECD is carried out in order to evaluate the permanent magnets temperature. It is shown that not only can the proposed device provide a high damping density but also can perfectly transfer the thermal energy to the air surrounding the device.           57  5. Electromagnetic dampers with applications in structural engineering  5.1 Introduction In structural engineering the mitigation of damage induced by sever dynamic loads such as earthquakes and strong winds is an eminent interest. Over the past decades, finding an effective system to protect structures has been one of the major challenges for civil engineers. Among the various techniques, structural control methods show a great potential for reducing the damage effects of seismic excitations. Therefore, several control devices and algorithm have been investigated recently. In general, control systems can be categorized into four groups according to their operations. Passive devices, which require no external power, never destabilize structures. However, they have low adaptability to changes in external loads. On the other hand, active control devices are adaptive to the varying applied loads, but their stability and large power consumption are still major problems. Semi-active devices are the third group of structural control systems, in which mechanical properties such as stiffness can be modified to improve their operations. Similar to active devices, semi-active systems require control algorithms to adapt their performance in response to vibration loads. However, they require only a slight amount of energy. Semi-active systems only dissipate the energy, and do not input forces to structures to affect their stability. In the last group of control systems, active or semi-active devices are integrated with passive devices as a hybrid system, thereby offering the adaptability of active and semi-active devices combined with the features of passive devices.  Electromagnetic systems can operate as passive (Chapter 3 and 4), semi-active and active (Chapter 3) control devices. A newly developed passive eddy current dampers (Chapter 3) can  58  provide a high damping force, but their performances are not adjustable. To resolve the difficulties, a hybrid electromagnetic damper (HEMD) is proposed.  The present chapter deals with applications of electromagnetic systems for structural engineering. CBED and ECD prototypes are designed here. Moreover, the pros and cons of each system are briefly discussed and compared with a bench mark conventional viscous damper. In order to comprise the unique features of active, semi-active and passive dampers, the HEMD is proposed. Figure 5-1 illustrates the proposed HEMD. This configuration consists of two parts: external and internal. The CBED is considered as an internal part, surrounded by the developed ECD as an external part. Therefore, the internal part operates as an active or semi-active device while the external part performs as a passive damper.    59   Figure 5-1. Hybrid electromagnetic damper.  5.2 Coil-based electromagnetic damper The CBED can be considered as passive, semi-active and active control devices. The four phase optimum passive CBED was discussed in Chapter 3. Table 5-1 and Figure 5-2 illustrate the damper parameters.    60  Table 5-1. Optimum design parameters for the CBED. Parameters Description Value tm Magnet Length 40 mm ti Iron Pole Length 20 mm tc Coil Length 40 mm rm Magnet Radius 100 mm ra Air Gap Radius 101 mm rc Coil Radius 109.5 rs Damper Shell Radius 121 mm rw Wire Radius 21 AWG CVPEM Passive Damping Density 3,061 kN-s/m4 Nm Number of required Magnets 81 C Provided Damping Coefficient  906 kN-s/m    Figure 5-2. Four phase CBED.  For the purposes of comparison, the capacity of a viscous fluid damper that has been manufactured by Taylor Device and installed in San Bernardino Country Medical Center is chosen as a bench mark damper (Taylor, 2003). Table 5-2 compares the capacity of CBED with the conventional viscous damper assuming a linear force-velocity relationship. As shown, the optimum passive CBED can operate similar to regular oil dampers. However, when the CBED is considered as semi-active, active or energy harvester systems, their damping density will be  61  decreased due to the additional resistance or power which is imposed to the device circuit (Chapter 3). Table 5-2. Comparison between bench mark damper and CBED. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2746 CBED 906 0.3 3061  5.3 Eddy current damper The novel ECD which is classified as a passive control device was developed in Chapter 4. As stated, the proposed ECD can provide a high damping density as 2,733 kN-s/m4. Table 5-3 review the optimum design parameters for the ECD. Table 5-3. Optimum parameters for ECD. Parameters Description Value Ag Air Gap 1 mm tp Conductor Thickness 5 mm θ  (Np) Magnet Angle (Number of Conductive Plate ) 36o (10) R2 Outer Magnets Radius  125.5 mm R1 Inner Magnets Radius 11 mm Lp Conductor Plate Length 125.5 mm r0 Plates Intersection Radius 7 mm tm Thickness of Magnets in Z direction 23 mm  Nmr Number of Magnet Rows 291 CVEC Damping Density 2,733 kN-s/m4 C Provided Damping Coefficient  906 kN-s/m   62    Table 5-4 compares the novel ECD with both the bench mark oil damper and CBED. It can be inferred that the novel ECD can produce a damping force in the range of regular oil dampers. However, the passive control systems are not adaptable to the load patterns. For this reason, a HEMD is proposed in the next section with the aim of integrating the innovative features of both electromagnetic systems. Table 5-4. Comparison between CBED and ECD. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2,746 CBED 906 0.3 3,061 ECD 906 0.34 2,733  5.4 Hybrid electromagnetic damper The HEMD is investigated in this section. Figure 5-1 illustrates the HEMDs configuration. CBED and ECD are treated as internal and external parts which operate as two parallel systems. The total damping force produced by HEMD, FHEMD, is the sum of the damping force produced by each part:  HEMD GEMD ECDF F F         (5-1) The parameters described in Table 5-1 for the optimum four phase CBED are considered as the internal part, while the external part parameters, ECD, will be defined based on mathematical model developed in chapter 4. Table 5-5 defines the hybrid electromagnetic damper parameters.  63    Figure 5-3. A sample HEMD, (a). 3D view, (b). Cross section.  64  Table 5-5. HEMD parameters. Parameters Description Value tm CBED Magnet Length 40 ti Iron Pole Length 20 tc Coil Length 40 rm CBED Magnet Radius  100 mm ra Air Gap Radius 101 mm rc Coil Radius 109.5 rs CBED Shell Radius and Inner ECD Magnet Radius  121 mm rw Coil Wire Radius  21 AWG Nm Number of Magnets in CBED  44 Ag ECD Air Gap 1 mm tp  Conductor Plate Thickness 5 mm θ  (Np) Magnet Angle (Number of Conductive Plate ) 36 (10) tecm ECD Magnet Thickness 23 mm Nmr Number of Magnet Rows in ECD 153 CVHEM Damping Density 2,049 kN-s/m4 C Provided Damping Coefficient  906 kN-s/m  As Table 5-5 shows, the HEMD damping capacity is less than the ECD and its volume is 1.34 times greater than the bench mark damper. However, the HEMDs offer the controllable vibration systems which can be also utilized as a harvesting energy system.   65  Table 5-6. Comparison between developed dampers. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) San Bernardino 906 0.33 2,746 Passive CBED 906 0.3 3,061 ECD 906 0.34 2,733 HEMD Total 906 0.44 2,049 External Part 411  Internal Part 495   5.5 Conclusion The application of electromagnetic devices for structural systems has been investigated in the present study. CBED and ECD prototype are designed and compared with the viscous damper. The results show that it is feasible to employ the electromagnetic devices as a vibration control systems for the structural applications. However, in order to improve their performance, a HEMD consisting of CBED and ECD dampers is proposed and designed. The HEMD offers both a stable and adjustable damping force. The application of HEMDs can also be extended as the energy harvesting system.      66  6. Summary and conclusion  6.1 Conclusion  The present research deals with the concept of electromagnetic systems and their applications in Civil Engineering. Three different electromagnetic devices, Coil-Based Electromagnetic Damper (CBED), Eddy Current Damper (ECD) and Hybrid Electromagnetic Damper (HEMD) were investigated through mathematical and numerical models. The fundamental electromagnetic theories, Lorentz and Faraday’s laws, are utilized to formulate a mathematical model. Numerical simulations are conducted using the finite element program, Flux. Results show that both modeling approaches are consistent with each other. Finally, the verified analytical models and Genetic Algorithm (GA) are employed in order to examine the maximum damping capacity and their corresponding geometrical parameters. After a brief introduction on vibration control systems and the concept of electromagnetic theories in chapter 1 and chapter 2, CBEDs are investigated in chapter 3. CBEDs can operate as passive, semi-active and active systems. They can also be considered as an energy harvesting system that convert kinetic energy into electricity using an additional circuit attached to the damper. Conducting an optimization led to the optimum design of four phase passive CBED. However, the operation of CBEDs relies on the additional resistance attached to the damper. Therefore, when they are employed as a semi-active, active or an energy harvester, their damping capacity is sharply reduced.  An innovative ECD with high damping density is proposed in chapter 4. The arc segment permanent magnets with circumferential magnetization are used to produce a high strength magnetic field. A theoretical model of the proposed ECD is constructed using electromagnetic theory under quasi-static condition. The magnetic flux and eddy current damping force are  67  quantified analytically and validated using finite element simulation. Although the analytical model neglects the effects of the time-variant property of the eddy currents, the analytical model is still able to approximate the damping force of the proposed ECD accurately. The developed analytical model is used to optimize the ECD design using genetic algorithm. The result shows that the proposed ECD can produce a high damping density up to 2,733 kN-s/m4, which is feasible to be used for structural applications.  With the design of CBEDs, the idea of adding the ECDs is presented and developed as a HEMD. The HEMDs consist of two parts, the CBED forms the internal part, while the ECD is located externally. The present study indicates that coupling EDCs and CBEDs can produce adequate damping force. Furthermore, these devices can be considered as an energy harvesting system. Table 5-7 summarizes the capacity of the systems developed in the present research.  Table 5-7. A summary of the developed electromagnetic dampers. Damper Damping Coefficient (kN-s/m) Volume (m3) Damping Density (kN-s/m4) Passive CBED 906 0.3 3,061 ECD 906 0.34 2,733 HEMD Total 906 0.44 2,049 External Part 411  Internal Part 495    68  6.2 Suggestions for future research Based on the findings of the present study, the following research is recommended for the future: 1. The detailed analytical and numerical model were provided for the developed eddy current damper. It will be important to verify the theories through experimental investigations. 2. This study only compared the three types of electromagnetic system in the component level. To extend this study, the developed devices can be implemented in a real bench mark structure for the system level investigation. 3. In the developed eddy current damper, the magnetic field was produced using the arc segment permanent magnet. The coil system can be used instead of permanent magnets to generate the magnetic field. 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Smart Materials and Structures, Volume 10.                          73  Appendices  Appendix A: Analytical expression for magnetic potential  2 2 2( )1 1 1( , , ) ( 1) , ,4      i j km i j ki j kMr z R z       (A-1)           22 222 2, , 2 cos cos arctan2 cos cos                    i k j k i i j i jki i j i jr z z r R rR R rz zr R rR R r         (A-2)                            22 22 22 2 2 222 22 22 cos coscos arctan2 cos cos 2 coslog cos 2 coscos log 2                                          i i j i ji j ki i j i j i i j kk i j i i j ki j k ir R rR R rR r z zr R rR R r r R rR z zz z R r r R rR z zR r z z r R rR                 2cos     i j kz z          74  Appendix B: Details of finite element model  ################################################################################                                   GENERAL DATA                                   ################################################################################    Project name :                 | C:\Windows\Temp\3DConfiguration.FLU   Project dir size :             | 1543.41  Mbytes                                      Software version :             | Flux3D (11.2)                                        Application :                  | MagneticTransient3D                                  Last modification :            | 26/11/2015  00:41:47                                 Date of the report :           | 26/11/2015  00:45:12                               -------------------------------------------------------------------------------------    ========== checkGeometry ==========   The geometry has been modified during the solving process to take into account the mechanical sets The verification of confused points is not allowed. No superimposed points No abnormal lines   ========== checkMesh ==========     Volume elements :    List of poor quality elements :       26     364     415     676     826    1030    1303    1307    1595    1599     1687    1688    1690    1781    1802    1832    1834    1838    1901    1902     1904    1905    1906    1907    1908    1910    2013    2020    2022    2023     2024    2064    2073    2074    2075    2124    2125    2126    2127    2128     2129    2131    2132    2139    2212    2233    2241    2242    2251    2252     2253    2254    2255    2257    2259    2260    2261    2262    2311    2318     2323    2334    2361    2374    2378    2379    2380    2381    2382    2383     2384    2389    2391    2395    2467    2471    2517    2518    2522    2530     2532    2534    2565    2613    2656    2658    2682    2689    2694    2695     2704    2710    2711    2787    2797    2805    2825    2846    2856    2875  75      2877    2907    2950    2952    2968    3004    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97749   97777   97921   97922   97923    97930   97931   97935   97946   97947   97948   97951   97956   97957   98015    98030   98221   98254   98416   98422   98534   98535   98542   98633   98811    98895   99045   99056   99060   99073   99116   99119   99121   99126   99127    99129   99130   99132   99134   99176   99247   99254   99280   99282   99295    99296   99304   99305   99306   99362   99493   99593  100161  100166  100191   100198  100366  100648  100686  100696  100721  100731  100736  100779  100786   100810  101176  101252  101259  101299  101321  101499  101734  101753  101766   101770  101787  101791  101803  101848  101955  102037  102065  102115  102159   102317  102356  102666  102669  102929  103039  103146  103190  103203  103342   103343  103344  103353  103428  103498  103630  103872  103927  103942  103948   103956  103971  103994  104021  104043  104047  104066  104067  104148  104226   104265  104432  104436  104438  104514  104516  104535  104581  104667  104702   104704  104705  104739  104749  104771  104830  104876  105001  105216  105242   105243  105248  105304  105325  105445  105482  105668  105738  105746  105748   105763  105769  105772  105777  105841  105845  105849  105853  105860  105861   105863  105870  105876  105882  105884  105946  105961  106127  106131  106133   106143  106212  106227  106238  106248  106330  106405  106688  106875  107191   107356  107366  107386  107762  107975  108001  108003  108196  108222  108268  83    108274  108319  108333  108933  109164  109295  109303  109418  109450  109475   109507  109508  109697  109728  109748  109772  109832  110012  110084  110125   110250  110390  110417  110432  110460  110463  110512  110542  110603  110753   110762  110787  110797  111000  111011  111054  111102  111267  111275  111416   111464  111723  111762  111816  111835  111838  111842  111851  111864  111876   111889  112020  112276  112277  112296  112332  112503  112574  112586  112600   112851  112906  113012  113090  113108  113346  113396  113624  113719  113781   113954  114046  114050  114233  114241  114304  114357  114495  114503  114521   114526  114533  114536  114544  114557  114765  114897  114937  115134  115172   115187  115188  115242  115398  115545  115570  115594  115656  116004  116236   116388  116581  116618  116658  116704  116721  116867  117151  117155  117214   117255  117282  117288  117524  117615  117621  117705  117869  118091  118095   118110  118467  118469  118509  118578  118693  118820  118857  118868  118983   119001  119107  119164  119193  119226  119249  119294  119330  119442  119572   119586  119668  119728  119857  120329  120496  120598  120600  120986  120992   121019  121085  121198  121245  121261  121443  121571  121628  121641  121666   121799  121896  122258  122507  122630  122833  122851  122897  122923  123108   123135  123152  123164  123170  123212  123415  123453  123727  123950  124070   124260  124327  124400  124407  124423  124443  124629   Volume elements :    Number of elements not evaluated     : 0  %    Number of excellent quality elements : 13.02  %    Number of good quality elements      : 63.99  %    Number of average quality elements   : 21.02  %    Number of poor quality elements      : 1.96  %   Number of nodes : 244586 Number of line elements : 4518 Number of surface elements : 79851 Number of volume elements : 124713 Mesh order : 2nd order   Verifying of linked faces ...   89 face(s) respect strongly (ELEMENTS) linked mesh generator.    84  List of linked faces of which NODES do not verify linked mesh generator: 27  31  39  43  111  114  121  124   List of linked faces without source face (geometry): 119  129  167      ========== checkPhysic ==========   Begin of physical check ...   Multiplying coefficient for the flux in the coils and for coupling with electrical circuits : Automatically calculated value = 1   The mechanical set PLATES does not have an internal characteristic (inertia, frictions). The internal friction losses torque will not be computed End of physical check.   Multiplying coefficient for the flux in the coils and for coupling with electrical circuits : Automatically calculated value = 1   The mechanical set PLATES does not have an internal characteristic (inertia, frictions). The internal friction losses torque will not be computed    ========== checkCircuit ==========      ========== Loaded macros ==========   EXTRUDEFACEREGIONS AUTOMATICREPORT  ################################################################################                              GEOMETRICAL PROPERTIES                              ################################################################################  CoordSys   Num | Name    | Comment                                  | Type      | Parent      | Origin          | Rotations (deg)   --------------------------------------------------------------------------------------------------------------------------   1   | XYZ1    | Standard Cartesian system of coordinates | Cartesian | GlobalUnits | ['0', '0', '0'] | ['0', '0', '0']     2   | Z_ON_OY | Cartesian system with z axis on -OY      | Cartesian | GlobalUnits | ['0', '0', '0'] | ['90', '0', '0']   85    3   | Z_ON_OX | Cartesian system with z axis on OX.      | Cartesian | GlobalUnits | ['0', '0', '0'] | ['90', '90', '0']   4   | XY1     | Cartesian system of coordinates.         | Cartesian | GlobalUnits | ['0', '0', '0'] | ['0', '0', '0']   --------------------------------------------------------------------------------------------------------------------------  InfiniteBoxCylinderZ    size = ['150','180','400','450']  -------------------------------------------------------------------------------- DomainType3D    lengthUnit = LengthUnit['MILLIMETER']    angleUnit = AngleUnit['DEGREE']    INFINITE_EXP = 2.0  -------------------------------------------------------------------------------- TransfTranslationVector    name = 'MAGNET'    coordSys = CoordSys['XYZ1']    vector = ['0','0','50']  TransfTranslationVector    name = 'PLATE1'    coordSys = CoordSys['XYZ1']    vector = ['0','0','50']  TransfTranslationVector    name = 'PLATE2'    coordSys = CoordSys['XYZ1']    vector = ['0','0','-50']  PointCoordinates   Num | CoordSys | uvw                                                             | Nature   | Mesh               ------------------------------------------------------------------------------------------------------------------   1   | XY1      | ['0', '0', '0']                                                 | STANDARD | AIDED_MESHPOINT001   2   | XY1      | ['75', '0', '0']                                                | STANDARD | AIDED_MESHPOINT001   3   | XY1      | ['0', '2.375', '0']                                             | STANDARD | AIDED_MESHPOINT001   4   | XY1      | ['0', '-2.375', '0']                                            | STANDARD | AIDED_MESHPOINT001   5   | XY1      | ['75', '-2.375', '0']                                           | STANDARD | AIDED_MESHPOINT001   6   | XY1      | ['75', '2.375', '0']                                            | STANDARD | AIDED_MESHPOINT001   7   | XY1      | ['0', '3.175', '0']                                             | STANDARD | AIDED_MESHPOINT001  86    8   | XY1      | ['53.0330085889911', '56.2080085889911', '0']                   | STANDARD | AIDED_MESHPOINT001   9   | XY1      | ['75', '3.175', '0']                                            | STANDARD | AIDED_MESHPOINT001   10  | XY1      | ['0', '-3.175', '0.704991620637E-15']                           | STANDARD | AIDED_MESHPOINT001   11  | XY1      | ['75', '-3.175', '0.704991620637E-15']                          | STANDARD | AIDED_MESHPOINT001   12  | XY1      | ['53.0330085889911', '-56.2080085889911', '0.124806850608E-13'] | STANDARD | AIDED_MESHPOINT001 ------------------------------------------------------------------------------------------------------------------  PointPropagated   Num | Pt orig | Transf | Nature   | Mesh            -----------------------------------------------------   13  | 10      | MAGNET | STANDARD | AIDED_MESHPOINT   14  | 11      | MAGNET | STANDARD | AIDED_MESHPOINT   15  | 12      | MAGNET | STANDARD | AIDED_MESHPOINT   16  | 7       | MAGNET | STANDARD | AIDED_MESHPOINT   17  | 8       | MAGNET | STANDARD | AIDED_MESHPOINT   18  | 9       | MAGNET | STANDARD | AIDED_MESHPOINT   19  | 1       | PLATE1 | STANDARD | AIDED_MESHPOINT   20  | 3       | PLATE1 | STANDARD | AIDED_MESHPOINT   21  | 6       | PLATE1 | STANDARD | AIDED_MESHPOINT   22  | 2       | PLATE1 | STANDARD | AIDED_MESHPOINT   23  | 5       | PLATE1 | STANDARD | AIDED_MESHPOINT   24  | 4       | PLATE1 | STANDARD | AIDED_MESHPOINT   25  | 1       | PLATE2 | STANDARD | AIDED_MESHPOINT   26  | 3       | PLATE2 | STANDARD | AIDED_MESHPOINT   27  | 6       | PLATE2 | STANDARD | AIDED_MESHPOINT   28  | 2       | PLATE2 | STANDARD | AIDED_MESHPOINT   29  | 5       | PLATE2 | STANDARD | AIDED_MESHPOINT   30  | 4       | PLATE2 | STANDARD | AIDED_MESHPOINT   31  | 16      | MAGNET | STANDARD | AIDED_MESHPOINT   32  | 17      | MAGNET | STANDARD | AIDED_MESHPOINT   33  | 18      | MAGNET | STANDARD | AIDED_MESHPOINT   34  | 31      | MAGNET | STANDARD | AIDED_MESHPOINT   35  | 32      | MAGNET | STANDARD | AIDED_MESHPOINT   36  | 33      | MAGNET | STANDARD | AIDED_MESHPOINT   37  | 34      | MAGNET | STANDARD | AIDED_MESHPOINT   38  | 35      | MAGNET | STANDARD | AIDED_MESHPOINT   39  | 36      | MAGNET | STANDARD | AIDED_MESHPOINT   40  | 13      | MAGNET | STANDARD | AIDED_MESHPOINT   41  | 14      | MAGNET | STANDARD | AIDED_MESHPOINT   42  | 15      | MAGNET | STANDARD | AIDED_MESHPOINT   43  | 40      | MAGNET | STANDARD | AIDED_MESHPOINT   44  | 41      | MAGNET | STANDARD | AIDED_MESHPOINT   45  | 42      | MAGNET | STANDARD | AIDED_MESHPOINT   46  | 43      | MAGNET | STANDARD | AIDED_MESHPOINT   47  | 44      | MAGNET | STANDARD | AIDED_MESHPOINT   48  | 45      | MAGNET | STANDARD | AIDED_MESHPOINT   49  | 19      | MAGNET | STANDARD | AIDED_MESHPOINT   50  | 20      | MAGNET | STANDARD | AIDED_MESHPOINT  87    51  | 21      | MAGNET | STANDARD | AIDED_MESHPOINT   52  | 22      | MAGNET | STANDARD | AIDED_MESHPOINT   53  | 23      | MAGNET | STANDARD | AIDED_MESHPOINT   54  | 24      | MAGNET | STANDARD | AIDED_MESHPOINT   55  | 49      | MAGNET | STANDARD | AIDED_MESHPOINT   56  | 50      | MAGNET | STANDARD | AIDED_MESHPOINT   57  | 51      | MAGNET | STANDARD | AIDED_MESHPOINT   58  | 52      | MAGNET | STANDARD | AIDED_MESHPOINT   59  | 53      | MAGNET | STANDARD | AIDED_MESHPOINT   60  | 54      | MAGNET | STANDARD | AIDED_MESHPOINT   61  | 55      | MAGNET | STANDARD | AIDED_MESHPOINT   62  | 56      | MAGNET | STANDARD | AIDED_MESHPOINT   63  | 57      | MAGNET | STANDARD | AIDED_MESHPOINT   64  | 58      | MAGNET | STANDARD | AIDED_MESHPOINT   65  | 59      | MAGNET | STANDARD | AIDED_MESHPOINT   66  | 60      | MAGNET | STANDARD | AIDED_MESHPOINT   67  | 61      | MAGNET | STANDARD | AIDED_MESHPOINT   68  | 62      | MAGNET | STANDARD | AIDED_MESHPOINT   69  | 63      | MAGNET | STANDARD | AIDED_MESHPOINT   70  | 64      | MAGNET | STANDARD | AIDED_MESHPOINT   71  | 65      | MAGNET | STANDARD | AIDED_MESHPOINT   72  | 66      | MAGNET | STANDARD | AIDED_MESHPOINT -----------------------------------------------------  LineSegment   Num | def points | Nature   -----------------------------   1   | 1,3        | STANDARD   2   | 1,4        | STANDARD   3   | 4,5        | STANDARD   4   | 3,6        | STANDARD   5   | 5,2        | STANDARD   6   | 2,6        | STANDARD   7   | 7,8        | STANDARD   9   | 7,9        | STANDARD   10  | 10,11      | STANDARD   12  | 10,12      | STANDARD   138 | 73,81      | STANDARD   140 | 75,83      | STANDARD   142 | 77,85      | STANDARD   144 | 79,87      | STANDARD   145 | 89,73      | STANDARD   146 | 91,75      | STANDARD   148 | 93,77      | STANDARD   150 | 95,79      | STANDARD   153 | 97,81      | STANDARD   154 | 99,83      | STANDARD   156 | 89,97      | STANDARD   157 | 101,85     | STANDARD   159 | 91,99      | STANDARD   160 | 103,87     | STANDARD   162 | 93,101     | STANDARD   164 | 95,103     | STANDARD -----------------------------  88   LineArc3PTS   Num | def points  | Nature   ------------------------------   133 | 73,74,75    | STANDARD   134 | 75,76,77    | STANDARD   135 | 77,78,79    | STANDARD   136 | 79,80,73    | STANDARD   137 | 81,82,83    | STANDARD   139 | 83,84,85    | STANDARD   141 | 85,86,87    | STANDARD   143 | 87,88,81    | STANDARD   147 | 89,90,91    | STANDARD   149 | 91,92,93    | STANDARD   151 | 93,94,95    | STANDARD   152 | 95,96,89    | STANDARD   155 | 97,98,99    | STANDARD   158 | 99,100,101  | STANDARD   161 | 101,102,103 | STANDARD   163 | 103,104,97  | STANDARD ------------------------------  LinePropagated   Num | Transf | Appli nb | Line orig | Nature   ------------------------------------------------   15  | MAGNET | 1        | 10        | STANDARD   17  | MAGNET | 1        | 11        | STANDARD   18  | MAGNET | 1        | 12        | STANDARD   21  | MAGNET | 1        | 7         | STANDARD   23  | MAGNET | 1        | 8         | STANDARD   24  | MAGNET | 1        | 9         | STANDARD   27  | PLATE1 | 1        | 1         | STANDARD   29  | PLATE1 | 1        | 4         | STANDARD   31  | PLATE1 | 1        | 6         | STANDARD   33  | PLATE1 | 1        | 5         | STANDARD   35  | PLATE1 | 1        | 3         | STANDARD   36  | PLATE1 | 1        | 2         | STANDARD   39  | PLATE2 | 1        | 1         | STANDARD   41  | PLATE2 | 1        | 4         | STANDARD   43  | PLATE2 | 1        | 6         | STANDARD   45  | PLATE2 | 1        | 5         | STANDARD   47  | PLATE2 | 1        | 3         | STANDARD   48  | PLATE2 | 1        | 2         | STANDARD   49  | MAGNET | 1        | 19        | STANDARD   50  | MAGNET | 1        | 20        | STANDARD   51  | MAGNET | 1        | 21        | STANDARD   52  | MAGNET | 1        | 22        | STANDARD   53  | MAGNET | 1        | 23        | STANDARD   54  | MAGNET | 1        | 24        | STANDARD   55  | MAGNET | 1        | 49        | STANDARD   56  | MAGNET | 1        | 50        | STANDARD   57  | MAGNET | 1        | 51        | STANDARD   58  | MAGNET | 1        | 52        | STANDARD   59  | MAGNET | 1        | 53        | STANDARD  89    60  | MAGNET | 1        | 54        | STANDARD   61  | MAGNET | 1        | 55        | STANDARD   62  | MAGNET | 1        | 56        | STANDARD   63  | MAGNET | 1        | 57        | STANDARD   64  | MAGNET | 1        | 58        | STANDARD   65  | MAGNET | 1        | 59        | STANDARD   66  | MAGNET | 1        | 60        | STANDARD   67  | MAGNET | 1        | 13        | STANDARD   68  | MAGNET | 1        | 14        | STANDARD   69  | MAGNET | 1        | 15        | STANDARD   70  | MAGNET | 1        | 16        | STANDARD   71  | MAGNET | 1        | 17        | STANDARD   72  | MAGNET | 1        | 18        | STANDARD   73  | MAGNET | 1        | 67        | STANDARD   74  | MAGNET | 1        | 68        | STANDARD   75  | MAGNET | 1        | 69        | STANDARD   76  | MAGNET | 1        | 70        | STANDARD   77  | MAGNET | 1        | 71        | STANDARD   78  | MAGNET | 1        | 72        | STANDARD   79  | MAGNET | 1        | 73        | STANDARD   80  | MAGNET | 1        | 74        | STANDARD   81  | MAGNET | 1        | 75        | STANDARD   82  | MAGNET | 1        | 76        | STANDARD   83  | MAGNET | 1        | 77        | STANDARD   84  | MAGNET | 1        | 78        | STANDARD   85  | MAGNET | 1        | 25        | STANDARD   86  | MAGNET | 1        | 26        | STANDARD   87  | MAGNET | 1        | 27        | STANDARD   88  | MAGNET | 1        | 28        | STANDARD   89  | MAGNET | 1        | 29        | STANDARD   90  | MAGNET | 1        | 30        | STANDARD   91  | MAGNET | 1        | 31        | STANDARD   92  | MAGNET | 1        | 32        | STANDARD   93  | MAGNET | 1        | 33        | STANDARD   94  | MAGNET | 1        | 34        | STANDARD   95  | MAGNET | 1        | 35        | STANDARD   96  | MAGNET | 1        | 36        | STANDARD   97  | MAGNET | 1        | 85        | STANDARD   98  | MAGNET | 1        | 86        | STANDARD   99  | MAGNET | 1        | 87        | STANDARD   100 | MAGNET | 1        | 88        | STANDARD   101 | MAGNET | 1        | 89        | STANDARD   102 | MAGNET | 1        | 90        | STANDARD   103 | MAGNET | 1        | 91        | STANDARD   104 | MAGNET | 1        | 92        | STANDARD   105 | MAGNET | 1        | 93        | STANDARD   106 | MAGNET | 1        | 94        | STANDARD   107 | MAGNET | 1        | 95        | STANDARD   108 | MAGNET | 1        | 96        | STANDARD   109 | MAGNET | 1        | 97        | STANDARD   110 | MAGNET | 1        | 98        | STANDARD   111 | MAGNET | 1        | 99        | STANDARD   112 | MAGNET | 1        | 100       | STANDARD   113 | MAGNET | 1        | 101       | STANDARD  90    114 | MAGNET | 1        | 102       | STANDARD   115 | MAGNET | 1        | 103       | STANDARD   116 | MAGNET | 1        | 104       | STANDARD   117 | MAGNET | 1        | 105       | STANDARD   118 | MAGNET | 1        | 106       | STANDARD   119 | MAGNET | 1        | 107       | STANDARD   120 | MAGNET | 1        | 108       | STANDARD   121 | MAGNET | 1        | 109       | STANDARD   122 | MAGNET | 1        | 110       | STANDARD   123 | MAGNET | 1        | 111       | STANDARD   124 | MAGNET | 1        | 112       | STANDARD   125 | MAGNET | 1        | 113       | STANDARD   126 | MAGNET | 1        | 114       | STANDARD   127 | MAGNET | 1        | 115       | STANDARD   128 | MAGNET | 1        | 116       | STANDARD   129 | MAGNET | 1        | 117       | STANDARD   130 | MAGNET | 1        | 118       | STANDARD   131 | MAGNET | 1        | 119       | STANDARD   132 | MAGNET | 1        | 120       | STANDARD ------------------------------------------------  LineExtruded   Num | Transf | option | point | Nature   ------------------------------------------   13  | MAGNET | 2      | 10    | STANDARD   14  | MAGNET | 2      | 11    | STANDARD   16  | MAGNET | 2      | 12    | STANDARD   19  | MAGNET | 2      | 7     | STANDARD   20  | MAGNET | 2      | 8     | STANDARD   22  | MAGNET | 2      | 9     | STANDARD   25  | PLATE1 | 2      | 1     | STANDARD   26  | PLATE1 | 2      | 3     | STANDARD   28  | PLATE1 | 2      | 6     | STANDARD   30  | PLATE1 | 2      | 2     | STANDARD   32  | PLATE1 | 2      | 5     | STANDARD   34  | PLATE1 | 2      | 4     | STANDARD   37  | PLATE2 | 2      | 1     | STANDARD   38  | PLATE2 | 2      | 3     | STANDARD   40  | PLATE2 | 2      | 6     | STANDARD   42  | PLATE2 | 2      | 2     | STANDARD   44  | PLATE2 | 2      | 5     | STANDARD   46  | PLATE2 | 2      | 4     | STANDARD ------------------------------------------            91  ################################################################################                                 MESH PROPERTIES                                  ################################################################################  MeshPoint   Num | Name               | Comment                 | Length unit | Value            | Color     -------------------------------------------------------------------------------------------------   1   | SMALL              | Small mesh size         | MILLIMETER  | 1.1875           | Yellow      2   | MEDIUM             | Medium mesh size        | MILLIMETER  | 2.98433409657833 | Turquoise   3   | LARGE              | Large mesh size         | MILLIMETER  | 7.50000000000001 | Red         4   | AIDED_MESHPOINT    | MeshPoint of aided mesh | MILLIMETER  | Dynamic()        | Cyan        5   | AIDED_MESHPOINT001 | MeshPoint of aided mesh | MILLIMETER  | Dynamic()        | Cyan      -------------------------------------------------------------------------------------------------  MeshLineRelativeDeviation   Num | Name              | Comment                | Color | Deviation value ----------------------------------------------------------------------------   1   | AIDED_MESHLINE    | Mesh line by deviation | Cyan  | 0.5               2   | AIDED_MESHLINE001 | Mesh line by deviation | Cyan  | 0.5             ----------------------------------------------------------------------------  MeshGeneratorAutomaticRelativeDeviation    name = 'AIDED_MESHGENERATOR : Automatic mesh by deviation'    relativeDeviation = 0.5  MeshGeneratorAutomaticInactiveDeviation    name = 'AUTOMATIC : Automatic mesh: triangles, tetraedra elements'  -------------------------------------------------------------------------------- MeshGeneratorNoMesh    name = 'NO_MESH : No mesh on the faces or volumes'  -------------------------------------------------------------------------------- MeshGeneratorMapped    name = 'MAPPED : Mapping mesh: rectangles, hexaedra elements'  -------------------------------------------------------------------------------- MeshGeneratorExtrusive  92     name = 'MeshGeneratorExtrusive_MAGNET'    transf = Transf['MAGNET']  MeshGeneratorExtrusive    name = 'MeshGeneratorExtrusive_PLATE1'    transf = Transf['PLATE1']  MeshGeneratorExtrusive    name = 'MeshGeneratorExtrusive_PLATE2'    transf = Transf['PLATE2']  -------------------------------------------------------------------------------- MeshGeneratorLinked    name = 'MeshGeneratorLinked_MAGNET'    transf = Transf['MAGNET']  ################################################################################                               PHYSICAL PROPERTIES                                ################################################################################  ApplicationPreprocessor3D      ApplicationMagneticTransient3D    formulationModel = MagneticTransient3DFormulationModelAutomatic       approximation = VectorPotentialApproximationEdge    scalarVariableOrder = ScalarVariableAutomaticOrder    vectorNodalVariableOrder = VectorNodalVariableAutomaticOrder    coilCoefficient = CoilCoefficientAutomatic    transientInitialization = TransientInitializationStaticComputation  -------------------------------------------------------------------------------- DomainType3D    lengthUnit = LengthUnit['MILLIMETER']    angleUnit = AngleUnit['DEGREE']    INFINITE_EXP = 2.0  -------------------------------------------------------------------------------- Material    name = 'MAGNET'    propertyBH = PropertyBhMagnetOneDirection       br = '1.2'       mur = '1.06'  Material    name = 'PLATE'    propertyBH = PropertyBhLinear       mur = '1'    propertyJE = PropertyJeLinear  93        rho = '1.724E-8'  -------------------------------------------------------------------------------- MechanicalSetFixed    name = 'MAGNETS'  MechanicalSetCompressibleRemeshing    name = 'AIR'  MechanicalSetTranslation1Axis    name = 'PLATES'    kinematics = LinearImposedPosition       Position = '0.02*Sin(2*Pi*TIME)'       initialVelocity = '0'    translationAxis = TranslationZAxis       coordSys = CoordSys['XYZ1']  RegionVolume    name = 'MAGNETS1'    magneticTransient3D = MagneticTransient3DVolumeMagnetic       material = Material['MAGNET']       formulation = Formulation['MT3SCA']    color = Color['Turquoise']    visibility = Visibility['INVISIBLE']    mechanicalSet = MechanicalSet['MAGNETS']  RegionVolume    name = 'PLATES'    magneticTransient3D = MagneticTransient3DVolumeSolidConductor       material = Material['PLATE']       formulation = Formulation['MT3TWOM']    color = Color['Turquoise']    visibility = Visibility['VISIBLE']    mechanicalSet = MechanicalSet['PLATES']  RegionVolume    name = 'MAGNETS2'    magneticTransient3D = MagneticTransient3DVolumeMagnetic       material = Material['MAGNET']       formulation = Formulation['MT3SCA']    color = Color['Turquoise']    visibility = Visibility['INVISIBLE']    mechanicalSet = MechanicalSet['MAGNETS']  RegionVolume    name = 'AIR'    magneticTransient3D = MagneticTransient3DVolumeVacuum       formulation = Formulation['MT3SCA']    color = Color['Turquoise']    visibility = Visibility['INVISIBLE']    mechanicalSet = MechanicalSet['AIR']  RegionVolume  94    Num | Name     | Color     | Type                                    | Mechanical set ---------------------------------------------------------------------------------------   1   | MAGNETS1 | Turquoise | MagneticTransient3DVolumeMagnetic       | MAGNETS          2   | PLATES   | Turquoise | MagneticTransient3DVolumeSolidConductor | PLATES           3   | MAGNETS2 | Turquoise | MagneticTransient3DVolumeMagnetic       | MAGNETS          4   | AIR      | Turquoise | MagneticTransient3DVolumeVacuum         | AIR            ---------------------------------------------------------------------------------------  -------------------------------------------------------------------------------- ################################################################################                                SPATIAL PROPERTIES                                ################################################################################  VariationParameterPilot   Num | Name          | Comment                                  | Ref. value -----------------------------------------------------------------------------   1   | TIME          | Time (s)                                 | 0.0          2   | LINPOS_PLATES | Linear pos. (m) of mechanical set PLATES | 0.0        -----------------------------------------------------------------------------   ################################################################################                                SOLVING PROPERTIES                                ################################################################################  Scenario    name = 'SCENARIO_1'    pilots = [MultiValues       parameter = VariationParameter['TIME']       intervals = [IntervalStepValue          minValue = 0.0          maxValue = 1.1          stepValue = 0.05]]    adaptive = InactivatedAdaptive  -------------------------------------------------------------------------------- SolvingOptions    parametersLinearSystemSolvers = ParametersLinearSystemSolvers  95        linearSystemSolverChoice = LinearSystemSolverAutomatic       parametersIccg = ParametersIccgAutomatic       parametersGmres = ParametersGmresAutomatic       parametersBicgstab = ParametersBicgstabAutomatic       parametersSuperlu = ParametersSuperluAutomatic       parametersMumps = ParametersMumpsAutomatic       parametersPardiso = ParametersPardisoAutomatic       parametersAutomaticScaling = ParametersAutomaticScalingAutomatic       userSubroutineLinearSolverParameters = ParametersLinearSolverByUserSubroutine       preLinearSolverChoice = PreLinearSolverAutomatic    newtonRaphsonParameters = ParametersNewtonRaphson       precision = 1.0E-4       maximumIterationNumber = 100       relaxationFactorComputationMethod = ParametersNewtonRaphsonRelaxationFactorComputationMethodAutomatic    electromagneticThermalCouplingParameters = ParametersElectromagneticThermalCoupling       updatingPrecision = 0.01       minAndMaxUpdatingNumbers = [1,5]       updatingsNonConvergenceBehavior = UpdatingsNonConvergenceContinue    initialTemperature = 293.15    t0ComputationMethod = T0ComputationMethodAutomatic    transientVariableSetToZeroType = TransientVariablesResettingAutomatic    automaticSaving = autoFrequencySaving    rescalingFactors = RescalingFactors       i1RescalingFactor = 1.0       i2RescalingFactor = 1.0       u1RescalingFactor = 0.001       v1eitRescalingFactor = 1.0       v2eitRescalingFactor = 1.0       t1wRescalingFactor = 1.0       t2wRescalingFactor = 1.0    edgeElementGaugeMethod = EdgeElementGaugeMethodAutomatic    MatMethodConstruction = BuildMatrixMethodAutomaticDetermination    symFormulations = autoFormulations    theta = 1.0    hj1ndSplittingMethod = Hj1ndComputationAutomaticSplittingDetermination    nonlinearZsComputationMethod = NonlinearZsComputationMethod1DFEModel    heElementNumber = 0  -------------------------------------------------------------------------------- AdaptiveSolverOptions    energyErrorThreshold = MediumThresholdAdaptiveSolver    iterationsMaxNumber = 4      

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