Design Sensitivity Analysis and Optimization of Electromagnetic Systems

1. Introduction. 1.1 Optimal Design Process. 1.2 Design Steps of Electromagnetic System. 1.3 Design Variables. 1.4 Equations and Characteristics of Electromagnetic Systems. 1.4.1 Maxwell's Equations and Governing Equations. 1.4.2 Characteristics of Electromagnetic Systems. 1.5 Design Sensitivity Analysis. 1.5.1 Finite Difference Method. 1.5.2 Discrete Method. 1.5.3 Continuum Method.



2. Variational Formulation of Electromagnetic Systems. 2.1 Variational Formulation of Electrostatic System. 2.1.1 Differential State Equation. 2.1.2 Variational State Equation. 2.2 Variational Formulation of Magnetostatic System. 2.2.1 Differential State Equation. 2.2.2 Variational State Equation. 2.3 Variational Formulation of Eddy Current System. 2.3.1 Differential State Equation. 2.3.2 Variational State Equation. 2.4 Variational Formulation of DC Conductor System. 2.4.1 Differential State Equation. 2.4.2 Variational State Equation.



3. Continuum Shape Design Sensitivity of Electrostatic System. 3.1 Material Derivative and Formula. 3.1.1 Material Derivative. 3.1.2 Material Derivative Formula. 3.2 Shape Sensitivity of Outer Boundary. 3.2.1 Problem Definition and Objective Function. 3.2.2 Lagrange Multiplier Method for Sensitivity Derivation. 3.2.3 Adjoint Variable Method for Sensitivity Analysis. 3.2.4 Boundary Expression of Shape Sensitivity. 3.2.5 Analytical Example. 3.2.6 Numerical Examples. 3.3 Shape Sensitivity of Outer Boundary for System Energy. 3.3.1 Problem Definition. 3.3.2 Lagrange Multiplier Method for Energy Sensitivity. 3.3.3 Adjoint Variable Method for Sensitivity Analysis. 3.3.4 Boundary Expression of Shape Sensitivity. 3.3.5 Source Condition and Capacitance Sensitivity. 3.3.6 Analytical Example. 3.3.7 Numerical Examples. 3.4 Shape Sensitivity of Interface. 3.4.1 Problem Definition and Objective Function. 3.4.2 Lagrange Multiplier Method for Sensitivity Derivation. 3.4.3 Adjoint Variable Method for Sensitivity Analysis. 3.4.4 Boundary Expression of Shape Sensitivity. 3.4.5 Analytical Example. 3.4.6 Numerical Example. 3.5 Shape Sensitivity of Interface for System Energy. 3.5.1 Problem Definition. 3.5.2 Lagrange Multiplier Method for Energy Sensitivity. 3.5.3 Adjoint Variable Method for Sensitivity Analysis. 3.5.4 Boundary Expression of Shape Sensitivity. 3.5.5 Source Condition and Capacitance Sensitivity. 3.5.6 Analytical Example. 3.5.7 Numerical Examples.







4. Continuum Shape Design Sensitivity of Magnetostatic System. 4.1 Interface Shape Sensitivity. 4.1.1 Problem Definition and Objective Function. 4.1.2 Lagrange Multiplier Method for Sensitivity Derivation. 4.1.3 Adjoint Variable Method for Sensitivity Analysis. 4.1.4 Boundary Expression of Shape Sensitivity. 4.1.5 Interface Problems. 4.1.6 Analytical Example. 4.1.7 Numerical Examples. 4.2 Interface Shape Sensitivity for System Energy. 4.2.1 Problem Definition. 4.2.2 Lagrange Multiplier Method for Energy Sensitivity. 4.2.3 Adjoint Variable Method for Sensitivity Analysis. 4.2.4 Boundary Expression of Shape Sensitivity. 4.2.5 Interface Problems. 4.2.6 Source Condition and Inductance Sensitivity. 4.2.7 Analytical Examples. 4.2.8 Numerical Examples.







5. Continuum Shape Design Sensitivity of Eddy Current System. 5.1 Interface Shape Sensitivity. 5.1.1 Problem Definition and Objective Function. 5.1.2 Lagrange Multiplier Method for Sensitivity Derivation. 5.1.3 Adjoint Variable Method for Sensitivity Analysis. 5.1.4 Boundary Expression of Shape Sensitivity. 5.1.5 Interface Problems. 5.1.6 Numerical Examples. 5.2 Interface Shape Sensitivity for System Power. 5.2.1 Problem Definition. 5.2.2 Adjoint Variable Method for Power Sensitivity. 5.2.3 Boundary Expression of Shape Sensitivity. 5.2.4 Sensitivities of Resistance and Inductance. 5.2.5 Numerical Examples.







6. Continuum Shape Design Sensitivity of DC Conductor System. 6.1 Shape Sensitivity of Outer Boundary. 6.1.1 Problem Definition and Objective Function. 6.1.2 Lagrange Multiplier Method for Sensitivity Derivation. 6.1.3 Adjoint Variable Method for Sensitivity Analysis. 6.1.4 Boundary Expression of Shape Sensitivity. 6.2 Shape Sensitivity of Outer Boundary for Joule loss power. 6.2.1 Problem Definition. 6.2.2 Boundary Expression of Shape Sensitivity. 6.2.3 Resistance Sensitivity. 6.2.4 Analytical Examples. 6.2.5 Numerical Examples.







7. Level Set Method and Continuum Sensitivity. 7.1 Level Set Method. 7.2 Coupling of Continuum Sensitivity and Level Set Method. 7.3 Numerical Considerations.







8. Hole and Dot Sensitivity for Topology Optimization. 8.1 Hole Sensitivity. 8.1.1 Hole Sensitivity in Dielectric Material. 8.1.2 Hole Sensitivity in Magnetic Material. 8.1.3 Numerical Examples. 8.2 Dot Sensitivity. 8.2.1 Dot Sensitivity in Dielectric Material. 8.2.2 Dot Sensitivity in Magnetic Material. 8.2.3 Numerical Examples.





Appendix A. More Examples of Electrostatic System. A.1 Outer Boundary Design. A.2 Outer Boundary Design for System Energy. A.3 Interface Design. A.4 Interface Design for System Energy.



Appendix B. More Examples of Magnetostatic System. B.1 Interface Design. B.2 Interface Design for System Energy.



Appendix C. More Examples of Eddy Current System. C.1 Interface Design for System Power.





Appendix D. More Examples of DC Conductor System. D.1 Outer Boundary Design for Joule Loss Power.