Rose-Hulman Institute of Technology Rose-Hulman Scholar Graduate Theses - Mechanical Engineering Graduate Theses Spring 2-2016 Implementation of Space-Time Finite Element Formulation in Elastodynamics Sidharth Ramesh Rose-Hulman Institute of Technology, rameshs1@rose-hulman.edu Follow this and additional works at: http://scholar.edu/ mechanical_engineering_grad_theses Part of the Other Mechanical Engineering Commons Recommended Citation Ramesh, Sidharth, "Implementation of Space-Time Finite Element Formulation in Elastodynamics" (2016). Graduate Theses - Mechanical Engineering. This Thesis is brought to you for free and open access by the Graduate Theses at Rose-Hulman Scholar. It has been accepted for inclusion in Graduate Theses - Mechanical Engineering by an authorized administrator of Rose-Hulman Scholar.
For more information, please contact bernier@rose- hulman. Implementation of Space-Time Finite Element Formulation in Elastodynamics A thesis Submitted to the Faculty Of Rose-Hulman Institute of Technology By Sidharth Ramesh In partial fulfilment of the requirements for the degree Of Masters of Science in Mechanical Engineering February 2016 © Sidharth Ramesh ROSE-HULMAN INSTITUTE OF TECHNOLOGY Final Examination Report Sidharth Ramesh Mechanical Engineering Name Graduate Major Thesis Title ____________________________________________________ Implementation of Space-Time Finite Element Formulation in Elastodynamics ______________________________________________________________ DATE OF EXAM: January 21, 2016 EXAMINATION COMMITTEE: Thesis Advisory Committee Department Thesis Advisor: Simon Jones ME Lorraine Olson ME Allen Holder MA X PASSED ___________ FAILED ___________ Abstract Ramesh, Sidharth 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 Space- Time 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.
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. ii Table of Contents List of Figures.
iv List of Abbreviations. v List of Symbols .1 Background on the Galerkin Approach .2 Finite Element Method .3 The Galerkin Finite Element Method in One Dimension .6 Isoparametric Shape Functions. Time Formulation in Single Degree of Freedom .3 Time Finite Element Method. Space Time Formulation for an axially vibrating bar .2 Axially Vibrating Bar.
60 iv List of Figures Figure 3.1 – One Dimensional Bar Figure 3.2 – Discretized Bar Figure 3.3 – The linear Element Figure 3.4 – Isoparametric Linear Element Figure 3.5 - Isoparametric Quadratic Element Figure 4.1 – Mass on Spring Single Degree of Freedom Model Figure 4.2 – Graphical depiction of the spring damper model Figure 4.2 – The response graph for the forced damped single degree of freedom model Figure 4.3 – Comparison between total time taken for each method vs time step Figure 4.4 – Comparison between Error vs Total Time Figure 4.5 – Comparison between time steps and relative error Figure 5.1 – 2 Dimensional Bar Figure 5.2 – Discretized Bar in Space and Time Figure 5.3 – Initial and Boundary conditions applied on the Bar Figure 5.3 – Axially vibrating bar for length=1m and height=.4 – Response graph for the bar Figure 5.5 - 3D plot of converged central difference solution Figure 5.6 - Comparison plot between relative change in error and space elements Figure 5.7 - 3D plot of central difference and space time methods Figure 5.8 – Comparison graph between relative difference and space elements Figure 5.9 – 3D plot of the space-time method’s computation time Figure 5.10 –3D plot of the Central Difference method’s computation time Figure E.1 – Nodal and Element locations in the discretized bar v List of Abbreviations FEM-Finite Element Methods TFEM-Time Finite Element Methods FDM-Finite Difference Methods STFEM- Space Time Finite Element Methods GM- Galerkin Method CD – Central Differences PDE-Partial Differential Equation vi List of Tables Tab 4.1 – Model properties of the mass-spring-damper single degree of freedom Tab 5.1 – Model properties of the axially vibrating bar vii List of Symbols x – Displacement ũ – Trial function - weighting function X - Ratio of element size to the total length of the bar N - shape function L - Length of bar c - damping coefficient q - Displacement corresponding to the time step q - Velocity q - Acceleration x - Difference between subsequent displacement values t - Difference between subsequent time values k - Stiffness v0 - Initial velocity x0 - Initial displacement r , S , T - Auxiliary coordinates - Density n - Natural frequency d - Damping frequency - Analytic damping coefficient C1 , C2 , C3 , C4 - analytic constant 1 1. Introduction Numerical techniques have been implemented to solve many engineering problems successfully in the past decades. Techniques exist with varying complexities for numerous problems, and the search for new, computationally efficient techniques has resulted in innovative formulations. This study investigates the applicability of the Space-Time finite element formulation to solve problems related to the field of elastodynamics.
Chapter 2 reviews current literature on the subject. The workings of the Galerkin method are explained in Chapter 3, the concept of time discretization using finite elements for a single degree of freedom case is explored in Chapter 4, and finally Chapter 5 looks at space-time finite element discretization for an axially vibrating bar. The computed results are presented and compared with existing techniques, and we conclude in Chapter 6. Implementing time discretization using finite elements has been of interest in the numerical computing society since 1987[3].
Accommodating time discretization along with space discretization in a single element formulation might result in greater accuracy and reduced computational time, especially in cases involving a specific time parameter, such as transient problems and dynamics. Many industries would benefit if computational efficiencies are improved, including the aero-engines industry. The following section introduces the academic framework and theoretical formulation of the Space-Time Finite Element Method (STFEM). Literature Review This chapter reviews the literature in the field of Space-Time Finite Element Methods (STFEM).
To provide a fundamental understanding, the following sections present the development of the academic literature from the basic ideas to the present topic.1 Introduction The branch of physics that tries to understand the behaviour of continuous media due to forces and their respective displacements is called continuum mechanics [1]. This specific field has been developed as a result of the amalgamation of the two broad fields: solid mechanics and fluid mechanics [1]. Some areas with specific emphasis have been developed with respect to the nature of the applied force and the behaviour of solids with respect to these forces (e. Other areas have been developed with respect to the nature of displacement of the material (e.
elastic and inelastic). The study of elastic solid behaviour under the influence of dynamic forces is the field of elastodynamics [2]. Broadly speaking, this field deals with the forces that cause the displacement of the medium to be in the form of waves. Problems like the impact of a rigid bar fixed to a wall, gradual force applied onto a spring and an impact force applied onto a spring where the displacements are in the elastic nature are dealt in the linear elastodynamics realm [2].
Many methods exist to solve elastodynamics problem, with Finite Element Methods (FEM) being one of the widely used numerical techniques. Innovative and sophisticated formulations have been developed and tested with the aim of improving accuracy and reducing computational time. The space-time finite element formulation is one such method [4].Scharpf [3] in their seminal paper titled ‘Finite Elements in Time and Space’, put forth the concept of finite element discretization for time dependent phenomena. They introduced the theoretical formulation of time discretization and elucidated the nature of time discretization.
They state the idea of finite element in time discretization and explain how it translates to fixed time interval. They illustrate their concept through a unidirectional bar example and then extend it to multi-degrees of freedom. Hughes and Gregory M. Hulbert [4], in their pivotal paper ‘Space-Time Finite Element Methods for Elastodynamics: Formulations and Error Estimates’, systematise and formalise the time discretization idea into the space-time finite element method and apply it to classical elastodynamic problems.
They present their analysis of the semi-discrete approach wherein space is first discretized using a finite element method and then time is discretized using a Finite Difference Method (FDM). They argue that it would be more efficient if time discretization is done using an FEM; STFEM could be used to circumvent the use of finite differences to develop systems of ordinary differential equations. They then explore a time discontinuous Galerkin formulation, which was developed for hyperbolic problems, and apply it to an elastodynamic example and present theoretical convergence analysis. Their formulation is used in various works either directly or in higher order approximations.
In another paper titled ‘Space-Time Finite Element Formulation for Second Order Hyperbolic Equations’ [5], they develop STFEM to be unconditionally stable and higher order accurate, in that orders of approximation higher than cubic degree can be used. They also point to advantages of the method like mathematically proving stability and convergence and extend the method to elastodynamics and higher order hyperbolic problems. French [6] takes the STFEM idea and applies it to the wave equation. In his work titled ‘A space-time finite element method for the wave equation’ he introduces his 4 formulation and compares it with that of Thomas J.
Hughes and Gregory M Hulbert.