Implementation of Space-Time Finite Element Formulation in Elastodynamics

Ramesh, Sidharth M.S.M.E Rose-Hulman Institute of Technology Feb 2016 Implementation of Space-Time Finite Element Formulation in Elastodynamics Thesis Advisor: Dr. Simon Jones Elastodynamics is an academic field that is involved in solving problems related to the field of wave propagation in continuous solid medium. Finite element methods have long been an accepted way of solving elastodynamics problems in the spatial dimension. Considerable thought has been given to ways of implementing finite element discretization in the temporal dimension as well. A particular method of finite element solving called space-time finite element formulation is explored in this thesis, which is a relatively recent technique for discretization in spatial and temporal dimensions. The present thesis explores the implementation of the Space-Time finite element formulation in solving classical elastodynamics examples, such as the mass-on-spring for a single degree of freedom and for an axially vibrating bar with multiple degrees of freedom. The space-time formulation is compared with existing finite difference techniques, such as the central difference method, for computational expenditure and accuracy. In the mass-on-spring case, the central difference method and linear time finite elements yield relatively similar results, whereas quadratic time finite elements are more accurate but take more time computationally. In the axially vibrating bar case, central difference is computationally more efficient than the SpaceTime finite element method. The final section concludes our findings and critiques the numerical effectiveness of the space-time finite element formulation. Dedication To my Parents for wholeheartedly supporting my decision to pursue higher education in the U.S. and for their omnipresent hand of guidance and encouragement And to Prof Jones, for his teaching and mentoring. If not for him I would not have attempted Finite Element for my thesis topic. I find immense inspiration in his ideals of “perseverance” and “guilt”. Acknowledgements I thank the Rose-Hulman Institute of Technology for giving me an opportunity to pursue my higher education in Mechanical Engineering. I am indebted to Rose for the help and support that I have obtained throughout my Master’s education. I sincerely thank my advisor Professor Jones for his omnipresent hand of guidance. His expertise in the world of Finite Element has always inspired me to dive into the deeper depths of Finite Element in Mechanical Engineering. Professor Jones’s emphasis on conceptual and abstract thinking and his problem solving approach have expanded my thinking and changed my work habits. I deeply admire his work ethic and I seek to emulate it in my life. I thank Professor Olson for inspiring me to extend my understanding of finite element into the real world and for giving me a chance to experiment with mechanical design. I thank Professor Holder for his patient and prompt help. If not for him, I would not have been able to get along with understanding the indispensable skill of numerical computing and Matlab programming. I thank Karen DeGrange for her helpful counsel throughout my Graduate study at Rose. Finally, I thank Terri Gosnell, who has always been there for help and guidance on the Graduate Process at the Rose-Hulman Institute of Technology. I greatly appreciate her for her timely and supportive help.