FEM: Electrostatics Physical modeling Basic equations of physical quantities Constitutive equation (material law) Relation Characterize the material Relative permittivity CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Physical modeling (cont.) Dielectric material (insulator) No free electric charges available. Polarization of the dielectric material, when exposed by an electric field. Basic Paraelectrica (e.g., water) Nonpolar (noble gas, e.g., H2, O2) Movement of charge centers polar Ferroelectrica (e.g., barium-titanate) Pyroelectrica (e.g., tourmaline) Alignment of existing dipols CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Physical modeling (cont.) Dielectric material (insulator) Hysteretic behavior Dielectric tensor (anisotropic material) CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Physical modeling (cont.) Interface conditions: dielectric displacement Starting point: Relation: Divergence theorem Surface charge (As/m2) CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Physical modeling (cont.) Interface conditions: dielectric displacement Result: Interface relations: Interface conditions: electric field intensity Starting point: CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Physical modeling (cont.) Interface conditions: electric field intensity Integration over interface: Interface relations CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Deriving PDE Introduce scalar electric potential Defining PDE Strong form: CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Weak form Test function: Integration over computational domain Greens integral theorem (integration by parts) Weak form CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Approximation: Substitution: Shape function for node CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher FEM: Electrostatics Algebraic system: Contains Dirichlet contributions Dirichlet node handling: CAE of Mechatronic Systems I (2008/2009), Prof. Manfred Kaltenbacher