Solution of Partial Differential Equations by electrical analogy

In this paper, PDEs are modeled by an electrical equivalent circuit generated from the equations arising from the finite element method (FEM). This allows the solution of PDEs to be obtained through circuit simulation. Our approach yields the same answer as FEM since the underlying equations are identical. The additional time required for transformation into an electrical network is negligible since it is done element by element and has been shown to be so in our experiments. Our approach naturally permits the simulation of coupled systems, where the electrical/mechanical devices, whose behaviour is governed by PDEs, are connected together through an electrical circuit. Further, the approach also permits a wide variety of electrical techniques such as the hybrid methods using both currents and voltages as unknowns, and parallelization techniques such as multiport decomposition, for the solution of the problem. Simple test problems (electrostatic analysis of p–n junction diode and a heat transfer problem) are analyzed by our circuit simulator to show the validity of the proposed approach. We also show that the approach works well for nonlinear PDEs.

Research highlights▶ PDEs are modeled by an electrical equivalent circuit generated from the equations arising from the Finite Element Method (FEM). ▶ Our approach naturally permits the simulation of coupled systems, where the electrical/mechanical devices, whose behaviour is governed by PDEs, are connected together through an electrical circuit. ▶ The approach also permits a wide variety of electrical techniques for the solution of PDEs. ▶ The applicability of our approach has been shown to two important areas, namely device physics and MEMS.