Date of Award

Fall 2022

Document Type

Open Access Dissertation


Mechanical Engineering

First Advisor

Paul Ziehl


There is an increased interest in induction welding of carbon fiber reinforced thermoplastic composite laminates within the aerospace industry. Currently, optimal manufacturing process parameters (e.g., current, power, and frequency settings, coil movement speed, etc.) are determined through an experimental trial-and-error approach. To reduce the need for an experimental approach that can determine the manufacturing process parameters, which would be expensive in both time and money, accurate and computationally efficient numerical tools are desired. This research is focused on improving the existing numerical tool, WelDone, in regard to its speed and accuracy.

WelDone utilizes a hybrid element formulation that combines traditional finite element nodal values at the nodes with boundary integral equations at the boundary surfaces of the mesh. The formulation is hybrid, since the magnetic field in the conductive domain is represented with vector elements. The Biot-Savart law is used to determine the magnetic field generated by the induction coil on the boundary of the thermoplastic composite laminate. The magnetic field is related to the magnetic scalar potential through Green’s function. The Green’s function effectively maps the Dirichlet boundary condition resulting from the Biot-Savart law to a Neumann boundary condition, which is required to solve the weak formulation of Maxwell’s equations. ’Dirichlet-to-Neumann’ mapping allows the integral over the non-conductive domain, the air between coil and laminate, to be reduced to an integral over the boundary surface of the conductive domain, thereby, eliminating the need to discretize the non-conductive domain and improve computational efficiency.

This dissertation focuses on what is required to model curved panels. To this end, 8-node hexahedral elements are introduced in the finite element part of the tool. The introduction of an 8-node hexahedral element revealed a numerical instability when applying numerical integration to the aforementioned ’Dirichlet-to-Neumann’ mapping. The numerical instability occurred when numerically integrating a double surface integral containing the Green kernel and the normal derivative of the Green kernel. To cope with this numerical instability, the integration was solved, in part, analytically.

This dissertation assesses the accuracy of the model by comparing the results from the model to the results of previous sources, to, the literature, and to results found from commercial software. The individual algorithms and the implemented element are verified. When compared to the commercial software, the heating patterns are similar, however the magnitude of the ohmic losses are not the same between the two models.

For future development, the author recommends verifying the current code with real life experiments before the code is developed in other directions. If speeding up the code is the goal, a faster language can be employed or the development of a different element can be pursued.


© 2022, Florentius Johannes van Zanten