Integrated Computational Materials, Process and Product Engineering – IC-MPPE 2022

Induction hardening involves multiple physical problems from the electromagnetic, thermal, mechanical and metallurgical domain. These highly nonlinear coupled problems are usually solved by applying finite element methods (FEM) thanks to the available physics-based knowledge in the form of partial differential equations (PDEs). However, the material properties are coupled to the PDEs and they need to be known beforehand for an accurate FEM.

To estimate the thermo-physical material data, we inform a learning machine about the underlying physics described by known PDEs, a procedure called hybrid modeling. A learning machine based on artificial neural networks (ANNs) is employed to estimate the temperature response of an inductively heated sample. ANNs are the tool of choice when it comes to finding nonlinear relations between process parameters and the material response. The ANN is trained using measured temperature data T(r,t) (in time and space) while respecting PDEs describing the coupled electromagnetic-thermal problems with initial conditions (ICs) and boundary conditions (BCs). The thermo-physical properties, in this work thermal conductivity and specific heat, are considered as unknowns, which are embedded in the PDEs. The unknown material data are temperature dependent and are estimated by ANNs in an optimization problem. The optimization problem involves minimizing an objective function, which contains cost terms including temperature data fit loss, PDE residual loss, initial and boundary condition fit losses. Once the ANNs informed PDEs are optimized the material data incorporated in the PDEs are easily extracted by the nonlinear functions, which are approximated by optimized ANNs. We test and verify the approach for different samples (magnetic and non-magnetic) whose material data are already known (partially or totally) with high accuracy.