Lagrange Multipliers for Multi-physic Field-Circuit Coupling

The linear and passive devices with distributed arameters, are modeled as multi-port Hamiltonian (pH) systems with a finite number of ports, coupled to external structures ith lumped parameters. Appropriate boundary conditions (BC) for the Partial Differential Equations (PDEs) of several hysical fields inside devices are used. Originally, they are Electric Circuit Element (ECE) BC, but they are  eneralized here for multi-disciplinary domains, such as elastic solids, acoustic and thermal devices, having two scalar interaction variables: the current (integral of a 2-orm - “flow”) and voltage (integral of a 2-form - “effort”), associated to the input or output signals for each terminal. Their internal field is approximated by the Partitioned Finite Element Method (PFEM), using Lagrange multipliers to enforce the BC. An original Differential Algebraic Equation (DAE) port-Hamiltonian canonic form is generated in this manner for devices which may have ECE BC and Absorbing BC. The main objective is to extract the devices frequency characteristics (circuit unctions vs frequency). Since the existing theories do not cover these kinds of numerical methods, the proposed approach is an original contribution to model order duction and fast simulation of the coupled Multi-physics systems.