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Total Lagrangian Formulation for Large Deformation Modeling Using Uniform Background Mesh

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Title:
Total Lagrangian Formulation for Large Deformation Modeling Using Uniform Background Mesh
Creator:
Bhosale, Nikhil Vasantrao
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
KUMAR,ASHOK V
Committee Co-Chair:
SANKAR,BHAVANI V

Subjects

Subjects / Keywords:
finite
implicit
lagrangian
mesh
nonlinear

Notes

General Note:
The need for optimized structures, new materials and increased safety standards has increased the demand of nonlinear analysis in recent years. The finite element method is used to numerically compute stiffness and internal force matrices and the corresponding iterative problem is solved using the modified Newton-Raphson method. A typical finite element program to perform this analysis has three steps: a representative finite element model, the analysis of the model and the interpretation of results. A representative finite element model and the formulation of the applied loads/boundary conditions are key factors for a reliable and accurate response prediction of the model. Implicit Boundary Method uses a uniform background mesh for the finite element analysis and thus avoids the need for a conforming mesh. Mesh generation difficulties can be avoided when a background mesh rather than a mesh that conforms to the geometry is used for the analysis. The geometry is represented by equations and is independent of the mesh and is immersed in the background mesh. The solution to boundary value problems is approximated or piece-wise interpolated using the background mesh. The main challenge is in applying the boundary conditions because the boundaries may not have any nodes on them. Implicit boundary method has been used for linear static and dynamic analysis and has shown to be an effective approach for imposing boundary conditions but has never been applied to nonlinear problems. The main objective of this thesis is to extend implicit boundary method to large deformation nonlinear analysis using the Total Lagrangian formulation. The equations are solved using the widely used modified Newton-Raphson method with loads applied over many load steps. Several test examples are studied and compared with traditional finite element analysis software for verification.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
5/31/2018

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TOTAL LAGRANGIAN FORMULATION FOR LARGE DEFORMATION MODELING USING UNIFORM BACKGROUND MESH By NIKHIL BHOSALE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2016

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2016 Nikhil Bhosale

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To my parents

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4 ACKNOWLEDGMENTS Firstly, I would like to express my sincere gratitude to my advisor, Dr. Ashok V. Kumar for the continuous support of my thesis and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research a nd writing of this thesis. I could not have imagined having a better advisor and mentor for my thesis study. I would like to thank Dr. Bhavani Sankar for being a member of my supervisory committee It is my honor to have him in my committee and be guided f or my thesis. I am grateful for his willingness to review this thesis and provide valuable suggestions. Last but not the least, I would like to thank my family: my parents and to my brothers and sister for supporting me spiritually throughout writing this thesis and my life in general.

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5 TABLE OF CONTENTS page ACKNOWLEDGME NTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 Overview ................................ ................................ ................................ ................. 13 Goals and Objectives ................................ ................................ .............................. 15 Goal ................................ ................................ ................................ .................. 15 Objectives ................................ ................................ ................................ ......... 15 Outline ................................ ................................ ................................ .................... 16 2 INTRODUCTION TO IMPLICIT BOUNDARY METHOD ................................ ......... 17 Solution Stru cture ................................ ................................ ................................ ... 17 Imposing Essential Boundary Conditions ................................ ................................ 19 Essential Boundary Function ................................ ................................ .................. 19 Boundary Value Function ................................ ................................ ........................ 21 3 NONLINEAR FINITE ELEMENT ANALYSIS ................................ .......................... 24 Basic Principle ................................ ................................ ................................ ........ 24 Principle of Virtual Work ................................ ................................ .......................... 24 Total Lagrangian Formulation ................................ ................................ ................. 26 Linearization ................................ ................................ ................................ ........... 27 The Residual ................................ ................................ ................................ 27 The Newton Raphson Iteration ................................ ................................ ......... 28 Discretization ................................ ................................ ................................ .......... 29 4 IMPLICIT BOUNDARY METHOD FOR NONLINEAR ANALYSIS .......................... 33 Gover ning Equation for Nonlinear Analysis Using IBFEM ................................ ...... 33 Discretization Using IBFEM ................................ ................................ .................... 35 Constructing The Global Stiffness Matrix ................................ ................................ 36

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6 5 RESULT AND DISCUSSION ................................ ................................ .................. 38 2D Examples ................................ ................................ ................................ .......... 38 Plane Stress: A Plate Subjected to Uniform Pressure at the Top ..................... 38 Plane Stress: Thin Frame like Structure ................................ ........................... 41 Axisymmetric: Thin Disk ................................ ................................ ................... 43 3D Shell Example ................................ ................................ ................................ ... 46 Cantilever Subjected to End Shear Force ................................ ........................ 46 Application in Flexural Hinge Design ................................ ................................ ...... 48 6 CONCLUSION ................................ ................................ ................................ ........ 51 Summary ................................ ................................ ................................ ................ 51 Scope of Future Work ................................ ................................ ............................. 52 LIST OF REFERENCES ................................ ................................ ............................... 53 BIO GRAPHICAL SKETCH ................................ ................................ ............................ 55

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7 LIST OF TABLES Table page 5 1 Maximum deflection comparison. ................................ ................................ ....... 41 5 2 Maximum deflection comparison. ................................ ................................ ....... 43 5 3 Maximum deflection comparison. ................................ ................................ ....... 46 5 4 Vertical tip deflections for the cantilever loaded with end shear force. ............... 48

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8 LIST OF FIGURES Figure page 2 1 Representation of essential boundary and band in boundary element ............... 20 2 2 Boundary value function ................................ ................................ ..................... 22 3 1 Configuration at time 0, and ................................ ................................ ... 25 3 2 Configuration at time 0, and or 0, and iteration .......................... 28 5 1 A plane strain plate subjected to uniform pressure at the top ............................. 39 5 2 FE Model ................................ ................................ ................................ ............ 39 5 3 Displacement at P=100 ................................ ................................ ..................... 39 5 4 Stress along x direction ................................ ................................ ..................... 40 5 5 Von Mises Stress ................................ ................................ ............................... 40 5 6 Maxi mum Deflection vs Pressure ................................ ................................ .. 41 5 7 Thin beam structure ................................ ................................ ............................ 42 5 8 Thin beam structure ................................ ................................ ........................... 42 5 9 Maximum Deflection vs Pressure ................................ ................................ ... 43 5 10 Thin Disk ................................ ................................ ................................ ............ 44 5 11 FE Model ................................ ................................ ................................ ............ 44 5 12 Displacement at P=100 ................................ ................................ ...................... 45 5 13 Von Mises Stress ................................ ................................ .............................. 45 5 14 Maximum Deflection vs Pressure ................................ ................................ ... 45 5 15 Cantilever beam subjected to end shear load ................................ .................... 47 5 16 Cantilever beam model. ................................ ................................ ...................... 47 5 17 Deflection vs Load graph. ................................ ................................ ................... 47 5 18 Dimensions, Forces and coordinates of a flexural hinge ................................ .... 48 5 19 FE Model ................................ ................................ ................................ ............ 49

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9 5 20 Maximum D isplacement comparison ................................ ................................ .. 49 5 21 Von mises stress contours ................................ ................................ ................. 50

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10 LIST OF ABBREVIATIONS 3D Three Dimensions EBC Essential Boundary Condition FEM Finite Element Method IBFEM Implicit Boundary Finite Element Method NURBS Non Uniform Rational B Splines X FEM Extended Finite Element Method

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science TOTAL LAGRANGIAN FORMULATION FOR LARGE DEFORMATION MODELING USING UNIFORM BACKGROUND MESH By Nikhil Bhosale May 2016 Chair: Ashok V. Kumar Major: Mechanical Engineering The need for optimized structures, new materials and increased safety standards has increased the demand of nonlinear analysis in recent years. The finite element method is used to numerically compute stiffness and internal force matrices and the corresponding iterative problem is solved using the modified Newton Raphson method A typical finite element program to perform this analysis has three steps : a representative finite element model, the analysis of the model and the interpretation of results. A representative finite element model and the formu lation of the applied loads/boundary conditions are key factors for a reliable and accurate response pr ediction of the model. Implicit Boundary Method uses a uniform background mesh for the finite element analysis and thus avoids the need for a conforming mesh. Mesh generation difficulties can be avoided when a background mesh rather than a mesh that confo rms to the geometry is used for the analysis. The geometry is represented by equations and is independent of the mesh and is immersed in the background mesh. The solution to

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12 boundary value problems is approximated or piece wise interpolated using the back ground mesh. The main challenge is in applying the boundary conditions because the boundaries may not have any nodes on them. Implicit boundary method has been used for linear static and dynamic analysis and has shown to be an effective approach for imposi ng boundary conditions but has never been applied to nonlinear problems. The main objective of this thesis is to extend implicit boundary method to large deformation nonlinear analysis using the Total Lagrangian formulation. The equations are solved using the widely used modified Newton Raphson method with loads applied over many load steps. Several test examples are studied and compared with traditional finite element analysis software for verification.

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13 CHAPTER 1 INTR ODUCTION Overview When the deformation of the structure is large compared with its original configuration the response is considered nonlinear due to geometric nonlinearity even if the material behavior is still elastic. Such problems have been solved using two different formulations: Total Lagrangian (TL) and Updated Lagrangian (UL). In the former approach, the quantities of interest are mapped to the original undeformed configuration in the principle of virtual work. In the UL formulation, the geometr y is updated at the end of each load step. As the geometry is in the undeformed configuration for the TL formulation it is easier to implement and best suited for applications where the material is elastic. Large deformation analysis is needed for slender structures used in aerospace, civil and mechanical engineering applications. A variety of structures are designed to be compliant so that they undergo large deformation to facilitate the functioning of machinery and devices. In this thesis, we explore the possibility of using a background mesh for the analysis to avoid the difficulties related to mesh generation. The geometry is defined using equations of the boundaries and it is immersed in the background mesh. The boundaries can pass through the elements due to which we cannot assume that the nodes of the mesh are available at the boundaries for applying boundary conditions. The implicit boundary method was developed to apply essential boundary conditions on such boundaries and has been applied to a variet y of linear static and dynamic problems [1] [ 5 ]. In the present work, we study how this method could be extended to large deformation analysis.

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14 The traditional FEM uses a conforming mesh that approximates the geometry and provides the basis for piecewise interpolation of the field variables. But a conforming mesh is often difficult to generate, especially for complex geometry. As a result, many methods for avoiding mesh generation have been developed and other modifications of the traditional finite elemen t analysis have been developed that reduce the difficulty associated with mesh generation. These include meshless methods, isogeometric method, extended FEM and mesh independent methods that use a background mesh. An extensive overview of several meshless or meshfree methods can be found in several books and review papers such as Liu [ 6 ] and Gu [ 7 ]. Moving Least Square (MLS) method [ 8 ], Element Free Galerkin Method (EFGM) [ 9 ] and Meshless Local Petrov GalerKin Method [ 10 ] are examples of meshless methods. In the meshless methods, as set of nodes scattered over the domain is used for the analysis and these nodes are not connected to form elements. Meshless methods have also been applied to structural dynamics [1 1 ] [1 2 ]. Meshless methods use shape function th at do not need element connectivity but these shape functions are expensive to evaluate as a result. The geometry is approximated by the nodes on the boundary so applying boundary conditions is challenging and the basis functions used for meshless approxim ation do not have Kronecker delta properties. As a result other methods that are mesh independent such as XFEM have become more popular [1 3 ]. In XFEM the geometry of defects, such as cracks, are modeled using equations rather than the mesh itself so that t he mesh does not have to be modified to simulate crack propagation. In this thesis, we study a mesh independent approach where a background mesh is used for the analysis. This method has been referred to as the Implicit Boundary

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15 Finite Element Method (IBF EM) because we use the implicit boundary method to impose essential boundary conditions. A structured uniform background mesh that consists of regular shaped undistorted elements is used for interpolating or approximating the solution. Using a mesh with un distorted elements improves the quality of the solution by reducing numerical quadrature errors. Such a mesh is also easy to generate because it does not have to fit within the geometry. The bounding box within which the geometry fits is subdivided into un iform elements and then any element that is complete outside the geometry is removed to obtain the final background mesh for analysis. This process is easy to automate regardless of the complexity of the geometry. Furthermore, the implicit boundary method can be used with basis functions spline basis functions and meshless shape functions. Elements that use quadratic and cubic B splines have been developed. Application to static problems such as ela stic problems and heat conduction problems have been demonstrated in past work, but the feasibility of using IBFEM for non linear analysis has not been studied. Motivated by this, we explore the use of IBFEM for nonlinear analysis using Total lagrangian fo rmulation. Goals and Objectives Goal The main objective of this thesis is to extend the Implicit Boundary Finite Element Method (IBFEM) to large displacement non linear analysis using Total l agrangian formulation Several test examples are studied and com pared with traditional finite element analysis software for verification. Objectives The main objectives are listed below:

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16 1. Use the Implicit Boundary Finite Element approach to derive the linearized princip le of virtual work. 2. Discretize the equation for t he following cases: plane stress, plane strain, axisymmetric and 3D considering the solution structure for essential boundary condition. 3. Implement the discretized stiffness and load matrices for internal and boundary elements 4. Create examples to verify the numerical implementation. Outline The remaining chapters of this thesis are organized as follows: In chapter 2, the theory of IBFEM is explained. We introduce the basics of implicit boundary method. The equations for applying the essential boundary condi tions are also explained. In chapter 3 the theory of non linear finite element analysis is explained. Using the principal of virtual work and total lagrangian formulation the weak form is derived. Linearization and Discretization give us the tangent stiff ness and lo a d matrices. In chapter 4 the theory of IBFEM is applied to the formulation derived in chapter 3 The Linear strain displacement transformation matrix and Non Linear Strain Displacement Transformation matrix are derived considering the Dirichle t functions or the essential boundary functions. In chapter 5, we give some examples to validate the numerical implementation. The examples use 3D and 2D element types. The results of these examples are validated with the results of a traditional FEA software. In chapter 6, the summary of the work and conclusions are provided. The future work prospect is also given in this chapter.

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17 CHAPTER 2 INTRODUCTION TO IMPLICIT BOUNDARY METHOD A mesh, in a finite element analysis, is used for approximating the geometry of the structure that is being analyzed. It also represents the test and trial functions by piece wise interpolation. IBFEM uses a background mesh to avoid the need of a conforming mesh. The boundaries of the analysis domains are represented using implicit equations while a structured grid is used to interpolate functions. In a structured grid all the elements have regular geometry (squares/rectangles/cubes) and is much easier to generate compared to a conforming finite element me sh. The traditional methods used in FEM cannot be used for applying boundary conditions as the nodes are not guaranteed to be on the boundary. Implicit boundary method use s implicit equations for applying boundary conditions. In this chapter we introduce I BFEM and its solution structure. Solution Structure IBFEM is a mesh independent approach where a background mesh is used for analysis. The geometry of the analysis is represented using implicit equations which are also used to impose essential boundary con ditions. It uses a solution structure that ensures that the essential boundary conditions are imposed. Let u be a eld variable defined over or that must satisfy the boundary condition along a boundary which is part of the boundary of If ( and ) is the impl icit equation of the bou ndary then the solution structure for this field variable can be defined as ( 2 1 )

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18 This solution structure is then guaranteed to satisfy the condition along the boundary defined by the implicit equation for any The variable part of the solution structure is function This can be replaced by nite dimensional approximate function interpolation / approximation within elements of a structured grid Rvachev[ 14 ] developed the R functions as a way to construct the required implicit functions or characteristic functions Signed distance functions, originally made popular by t h e level set method, were also used to construct the characteristic function To satisfy prescribed boundary conditions solution structures consisting of R functions and distance functions were also used by Shapiro and Tsukanov [ 15 ] A highly nonlinear characteristic function over th e domain can To avoid these problems, the implicit boundary method uses approximate step functions as the characteristic fu nction so that over most of the domain this function has a unit value. Some advantages of this approach are that only the boundary elements are affected by the characteristic function, and if the mesh consists of uniform elements, then all the internal ele ments have identical stiffness matrix In traditional finite element mesh, the mesh conforms to the original geometry. This can generate distorted elements especially in case of complex geometries. These distorted elements are one of the causes of errors i n the solution. With a structured background mesh that does not conform to the geometry, inaccuracies arising due to distorted elements can be completely avoided because all the elements can have regular undistorted shapes. Furthermore, creating a backgrou nd mesh is easy and less time consuming as compared to creating a conforming mesh. For

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19 a standard part the step of creating a mesh can also be automated in case of structured background mesh. Imposing Essential Boundary C onditions To satisfy the essential boundary conditions or Dirichlet boundary conditions a trial solution structure can be constructed as: ( 2 2 ) In this solution structure, is a grid variable represented by piecewise approximation over the grid, and i s the boundary value function, that must be constru cted such that it has the specified boundary condition values at the boundary The variable part of the solution st ructure is homogeneous boundary conditions. is a diagonal matrix with components Di that are D functions that have a zero value on boundaries on which the th is the dimension of the problem. The test functions can also be constructed by using the D functions used for trial solution [1 5] ( 2 3 ) Essential Boundary F unction The essential boundary function or the dirichlet function is such that it vanishes on all boundaries where the th component of displacement is prescribed. D has a non zero gradient at these boundaries which ensures that the gradients of the displa cements are not constrained The dirichlet function should be non zero inside

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20 the analysis doma in. This ensures that the solution is not constrained anywhere with the domain of analysis. Thus, the dirichlet function must satisfy the following conditions ( 2 4 ) The dirichlet function is constructed using the implicit equation of the curve or surface representing the boundary. R functions have been used to construct Boolean combination of implicit functions [ 14, 16 ]. In this thesis, the approximate step function is constructed using implicit function of the boundary. Using the implicit equations used to define the boun dary conditions on a boundary, a step function at any point x is defined as follows: ( 2 5 ) Here D is equal to unity inside the solid as well as on the boundaries that do not have any Dirichlet boundary conditions specified. Figure 2 1. Representation of essential boundary and band in boundary element Essential boundary Band

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21 Within elements that contain a boundary with a Dirichlet boundary condition specified on the t h component of displacement, the approximate step function is constructed such that its value goes to zero at the boundary and to one, inside the geometry according to the step function defined in equation ( 2 7 ). Figure 2 1 shows an element on the bound ary of the domain and the band nea r the boundary within which the gradient of is non zero. Boundary Value F unction A boundary value function must be defined every time an essential boundary condition is imposed on a specific boundary. The value of must be equal to the imposed essential boundary condition. Consider a given situation where multiple essential boundary con dition might be specified at multiple boundaries. The resulting boundary value function should be a continuous function which gives you the imposed boundary conditions at respective boundaries and also transition smoothly. To get such a function a transfinite approach has been suggested [ 24 ] But this approach creates a rational function which is too non linear to be used in solution structure. A boundary value function can be constructed in numerous ways. One of the beneficial wa ys to construct a trial function is to use the shape functions used for the interpolation of grid variables. This ensures the trial function is a polynomial of the same order as the grid variable. A boundary value function can be constructed by piece wise interpolation of element shape functions. The following interpolation can be used for ( 2 6 )

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22 In Equation ( 2 8 ) are the shape functions of the grid element and are the nodal values of This interpolation is similar to th e grid variable If the and are not constructed using the same shape functions the solution structure will not be able to accurately represent constant strains. This is the main advantage of using the same element shape functions for defining and It also avoids the possibility of poor convergence which allows the solution structure to better approximate the exact solution. Figure 2 2 Boundary value function Thus, to obtain the desired boundary value function the nodes of the boundary elements should be assigned the nodal values Such nodal values are easy to obtain when the assigned values is constant or linearly varying along the boundary. The essential boundary function for the rest of the nodes that are not part of any boundary element can be set to zero. Consider the analysis domain in F igure 2 2 with the essential boundary condition The nodal values for all the nodes corresponding to the boundary element with essential boundary condition are set to while the rest Imposed essential boundary value =

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23 are set to zero. The boundary value function contributes to the load computation on the right side of weak form of governing equation

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24 CHAPTER 3 NONLINEAR FINITE ELEMENT ANALYSIS Basic P rinciple The governing finite element equations for a nonlinear analysis are derived using the same basic steps as in linear analysis: selecting interpolation functions and interpolation of element co ordinates and displacements with these functions in the governin g continuum mechanics equations. The derived finite element equations are then applied to every element. The total Lagrangian formulation method is derived using the weak form o r Principle of virtual work (PVW) in the underformed configuration The total applied load is divided into several time steps and Newton Raphson iterations are used to solve for the equilibrium. Due to the fact that the equation is expressed in the undeformed configuration it is easier to evaluate the volume surface integrals at eac h time step because the geometry is always in the original undeformed geometry and does not change at each time step. In this chapter, we discuss the total lagrangian approach is discussed and the corresponding matrix equations are derived Principle of V i rtual W ork To derive the finite element formulation we start from the weak form of the differential equations. From the standpoint of solid mechanics the weak form is the principle of virtual work. ( 3 1 )

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25 Here, represents the external virtual work or the work done by the external forces, is the Cauchy stress and is the virtual strain caused by the virtual displacement The external work done and the small strain is given as ( 3 2 ) ( 3 3 ) Where the superscripts and represent the boundary and surface forces. Figure 3 1. Configuration at time 0, and As shown in F igure ( 3 1 ) we need to calculate the configuration at time To get this configuration we can use two different formulations: Total Lagrangian (TL) and Updated Lagrangian (UL). In Total lagrangian formulation all integrals are calculated with respect to the initial undeformed configuration of the structure. In this thesis, we use the TL formulation as it is easier to implement and best suited for applications whe re material is elastic.

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26 Total Lagrangian Formulation The total Lagrangian formulation can be derived starting from the weak form or the Principle of Virtual Work (PVW) stated in a current configuration and transforming it back to the original undeformed c onfiguration [ 17 ]. In the process, the Cauchy stress is transformed into the second Piola Kirchoff stress and the small strain is transformed into the Green Lagrange strain. As the 2nd Piola Kirchoff stress tensor and the Green Lagrange strain tensor are energetically conjugate t he principle of virtual work can be stated as: ( 3 4 ) The left hand side represents the virtual strain energy expressed using index notation. is the second Piola Kirchoff stress tensor and is the virtual Lagrange strain both evaluated at any certai n time The volume of the domain is the original undeformed volume and the right hand side is the virtual work done by all the externally applied loads. The current configuration is ( 3 5 ) ( 3 6 ) where, is the original location of the point and is the total displacement up to the current configuration.

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27 Linearization The R esidual The left hand side of equation (3 4) is nonlinear in displacement so we need an iterative process to get to the final equilibrium solution. To linearize equation ( 3 4) we consider a residual ( 3 7 ) To get to the equilibrium equation we increment by ( 3 8 ) ( 3 9 ) So to get to the configuration at the iteration we add the deform ation to the current configuration ( 3 10 ) Now, if we consider the solution converges at iteration we have ( 3 11 ) Linearizing the above equation we get, ( 3 12 ) ( 3 13 ) Using the above equation we get the following Newton R aphson iter ation ( 3 14 )

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28 From the above equation we get deformation which is used to the update the current state ( 3 15 ) ( 3 16 ) Figure 3 2 Configuration at time 0, and or 0, and iteration The Newton Raphson Iteration The newton Raphson iteration can be given as ( 3 17 ) Applying the chain rule to the residual we get ( 3 18 ) Currently we are considering displacement independent load hence goes to zero. Therefore, ( 3 19 )

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29 ( 3 20 ) The above relation can be written as following for any iteration ( 3 21 ) Discretization The strain in index notation is given as ( 3 22 ) Using the above relation we get the relation for the change in virtual strain. This also gives us the discretized equation for the nonlinear contribution of strain to tangent stiffness ( 3 23 ) ( 3 24 ) For discretization we use the St. Venant Kirchoff material ( 3 25 ) The change in green strain is given as ( 3 26 ) Therefore using the above equations we get ( 3 27 )

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30 Similarly the Lagrange strain tensor due to virtual displacement can be given as ( 3 28 ) ( 3 29 ) This gives us the following discretized equation for the Newton Raphson iteration over all the elements in mesh as follows ( 3 30 ) Here, and are matrix and vector of second Piola Kirchoff stress The summation in the preceding equation is accomplished by assembling a tangen t stiffness matrix for the left hand side and a residual column matrix for the right hand side. The virtual displacement is canceled from both sides of the equations based on argument that the equation should be valid for any arbitrary virtual displacement This yields a global system of equations of the form: ( 3 31 ) Where, Linear part of the tangent stiffness matrix Nonl inear part of the tangent stiffness matrix Internal force vector The resultant externally applied load

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31 The definition of the strain displacement transformation matrices depends on the type of analysis one performs. Below we give only the matrices that apply to 2D problems but similar matrices are constructed for 3D as well. Using the definitions of incremen tal strains as described earlier we can write the following equations for a two dimensional element formulation: ( 3 32 ) ( 3 33 ) ( 3 34 ) Where, , , , and n is the number of nodes per element. The fourth row in the preceding matrices is only needed for axisymmetric problems. For 3D problems, three additional rows that correspond to extensional and shear strains in the direction must be added to these matrices. Similarl y, for the

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32 nonlinear part, we can define the matrices as follows for the axisymmetric case with the last row not needed for other 2D problems. ( 3 35 ) The global stiffness matrix and the global internal force vector is now constructed using the above discretized matrices. In the modified Newton Raphson method the global stiffness matrix is evaluated once every load step using the values of displacement components at the beginning of the load step iterations. The global force vector however, is revaluat ed at each iteration using the latest value of displacement.

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33 CHAPTER 4 IMPLICIT B OUNDARY METHOD FOR NONLINEAR ANALYSIS In this chapter we extend the theory of Implicit of Boundary method to non linear analysis. The major advantage of IBFEM is the use of structured background mesh instead of a conforming mesh. This potentially can elimi nate the difficulty of mesh generation that is required for any FEA analysis. Application of IBFEM to static problems such as elastic problems and heat conduction problems have been demonstrated in past work, in this chapter we extend it to nonlinear anal ysis. Governing Equation for Nonlinear Analysis U sing IBFEM To impose essential boundary conditions, in the implicit boundary method, the displacement and virtual displacement are expressed as: ( 4 1 ) ( 4 2 ) Her e is equal to unity inside the solid as well as on the boundaries that do not have any Dirichlet boundary conditions specified. Within elements that contain a boundary with a Dirichlet bound ary condition specified on the th component of displacement, is constructed as an approximate step function whose value goes to zero at the boundary as: ( 4 3 )

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34 Here is the distance function from the boundary of interest. Therefore at that boundary since the specified boundary condition is guaranteed to be enforced. Here is the boundary value function that must be constructed such that it has the specified boundary condition values at the boundary. is constructed by piece wise interpol ation or approximation using the shape functions of the element. The use of Dirichlet function changes the structure of the displacements assumed within each element that result in a modified shape function within the region where the step function transi tions from o ne to zero. In this transition region near the boundary, we have, ( 4 4 ) ( 4 5 ) ( 4 6 ) Where, is the modified shape function for the k th node. Note within the elements that do not have a boundary with displacement specified, therefore and which implies that all these elements are identical to the traditional finite elements except that some of the elements on the boundary may be partially inside and partially outside the domain of analysis. In these cases, the stiffness is computed accurate ly by only integrating over the region of the element that is within the domain. To do so, the element could be subdivided into triangles or tetrahedron so

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35 that the integration may be carried over only those triangles or tetrahedrons that are within the do main. Discretization U sing IBFEM In the traditional nonlinear finite element procedure [17 18 ], the preceding equation is discretized by deriving matrix equations that relate the quantities of interest with the nodal values of displacement or change in dis placement. As is already described in Chapter 3 this equation( 4 8 ) yields a global system of equations of the form: ( 4 7 ) To get the global tangent stiffness matrix and the internal force vector we define linear and nonlinear strain displacement matrices. Using the earlier definition of the strain displacement matrices equations ( 3 25 )( 3 26 )( 3 27 ) and the solution structure equations ( 4 1, 4 2 ) we get the following matrices, ( 4 8 ) ( 4 9 ) ( 4 10 )

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36 Where, , , , and n is the number of nodes per element. ( 4 11 ) In all the above relations we use the modified shape function Constructing T he Global S tiffness M atrix The construction of the global stiffness can therefore be implemented by using first t he unmodified shape function in equations ( 3 25 ) ( 27 ) for all the elements to construct element stiffness matrices and assembling them. Thereafter, the stiffness associated with the transition region where the step function transitions from one to zero can be computed as a line integral (for 2D and shell) or surface integrals (for 3D) as: ( 4 12 ) ( 4 13 ) These can be thought of as t he stiffness of the boundary and are also assembled in the global stiffness matrix. Note that the integration is first performed

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37 normal to the boundary with respect to the distance function and thereafter a line or surface integration is performed over the b oundary

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38 CHAPTER 5 RESULT AND DISCUSSION A few numerical examples are employed to verify the accuracy of non linear analysis using background mesh and Implicit Boundary Finite Element Method (or IBFEM). In the following example s we study the deformation of a cantilever subjected to end shear force [ 19 ] modeled using 3D shell elements. We also look at a plane stress and an axis symmetric example modeled using 2D elements and compare the results with commercial software. 2D Examples In this section, we will present examples using 2D elements. Plane Stress : A Plate Subjected to Uniform Pressure at the T op In this example a plate with a large width subjected to uniform pressure at the top is modelled as a plane stress example. Figure 5 1 shows the cross section with dimensions A uniform pressure is applied along the top surface of the structure and it is clamped at one end. The model with boundary conditions is shown in Fi gure 5 2. The material properties were s Poisson's ratio An initial pressure of is applied. The mesh used for the analysis using IBFEM is shown in figure 5 2 and it consists of 16 Node B Spline elements. The mesh density for this model is A model for the same structure was also created in ABAQUS which uses 164 quadratic quadrilateral elements of type CPS8R. For validation, the maximum di splacement, computed in the structure using IBFEM and Abaqus, is compared.

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39 In this example we also look at the stress plots for Von Mises and stress in radial direction. The stress contours obtained from IBFEM show good correlation with stress contours obt ained from Abaqus. Figure 5 1. A plane strain plate subjected to uniform pressure at the top Figure 5 2. FE Model F igure 5 3 Displacement at P=100 A). IBFEM B). Abaqus A B

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40 Figure 5 4 Stress along x direction A). IBFEM B). Abaqus Figure 5 5 Von Mises Stress A). IBFEM B). Abaqus B A A B

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41 Figure 5 6 Maximum Deflection vs Pressure A). IBFEM vs Abaqus B). Convergence of different mesh densities. Table 5 1 Maximum deflection comparison. P Max. deflection P Max. deflection IBFEM Abaqus IBFEM Abaqus 0 0 0 60 56.25 55.41 10 10.48 10.02 70 62.94 62.58 20 20.78 20.01 80 68.74 68.89 30 30.71 29.59 90 73.58 74.3 40 40.04 38.8 100 77.6 78.79 50 48.59 47.45 Plane Stress: Thin Frame like S tructure In this example a thin frame like structure is modelled using 2D plane stress elements. The str ucture is as shown in figure 5 7 This structure is rigidly clamped at both the ends and a uniform pressure is applied at the top. The model with boundary c onditions is shown in figure 5 8 (A). The material properties were s Poisson's ratio The structure has a cross section of An initial pressure of is applied. The mesh used for the analysis us ing IBFEM is shown in figure 5 8 ( A ) and it consists of 9 Node biquadratic elements. The mesh density for this model is A A B

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42 model for the same structure was also created in ABAQUS which use s 164 quadratic quadrilateral elements of type CPS8R. For validation, the maximum displacement, computed in the structure using IBFEM and Abaqus, is compared. As seen in the figure, the nonlinearity is mild for this problem but at large deformation the res ults from IBFEM and Abaqus are close and deviate significantly from the linear solution. Table 5 2 lists the maximum deflection of the structure and the corresponding applied pressure for the results obtained using IBFEM and Abaqus. Figure 5 7 Thin beam structure Figure 5 8 Thin beam structure A). FE model B). Structure Displacement A B

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43 Figure 5 9 Maximum Deflection vs Pressure A). IBFEM vs Abaqus B). Convergence study Table 5 2 Maximum deflection comparison. P Max. deflection P Max. deflection IBFEM Abaqus IBFEM Abaqus 100 3.640 3.702 260 10.111 9.973 120 4.403 4.472 280 10.971 10.760 140 5.178 5.249 300 11.850 11.650 160 5.969 6.031 320 12.740 12.340 180 6.773 6.816 340 13.660 13.130 200 7.583 7.603 360 14.851 13.920 220 8.410 8.393 380 15.510 15.310 240 9.248 9.182 400 16.480 15.500 Axisymmetric: Thin D isk In this example a thin disk like structure is modelled using 2D Axi symmetric elements. Figure 5 10 shows the cross section with dimensions and the revolved geometry of the structure. A uniform pressure is applied along the top surface of the disk and the disk is clamped along its edges. The model with boundary c onditions is shown in Figure 5 11 The ma terial properties were s Poisson's ratio An initial pressure of is applied. A B

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44 The mesh used for the analysis using IBFEM is shown in Figure 5 11 and it consists of 4 Node quadratic elements. The mesh density for the model is A model for the same structure was also created in SolidWorks. For validation, the maximum displacement, computed in the structure using IBFEM and SolidWorks, is compared. The re sults are also listed in Table 5 3. In this example we also look at the stress plots for Von Mises and stress in radial direction. The stress contours obtained from IBFEM show good correlation with stress contours obtained fr om Abaqus. Figure 5 10 Thin Disk A). Section B). Revolved geometry Figure 5 11 FE Model A B

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45 Figure 5 12 Displacement at P=100 Figure 5 13 Von Mises Stress A). IBFEM B). Abaqus Figure 5 14 Maximum Deflection vs Pressure A). IBFEM vs SolidWorks B). Convergence study A B

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46 Table 5 3 Maximum deflection comparison. P Max. deflection P Max. deflection IBFEM SW IBFEM SW 10 2.240 2.402 60 11.221 11.160 20 4.575 4.787 70 12.570 12.280 30 6.745 6.801 80 13.810 13.310 40 8.671 8.475 90 14.950 14.250 50 9.728 9.904 100 16.021 15.130 3D Shell E xample In this section, we will present examples using 3D elements. Cantilever Subjected to End Shear F orce In this example, we consider a thin cantilevered plate which is subjected to end shear force and is a benchmark problem that has been used in many studies [ 20 ]. The thin plate is illustrated in Figure 5 1 5 and its dimension are and The and the Poisson's ratio We use 3D shell element to model the plate and the displacement at the tip is used for the comparison and validation The background mesh used for this a nalysis is shown in the Figure 5 16 ( A ) which consists of 3D cubic B spline elements. The geometry is a surface which passes through these elements. The stiffness matrix of these elements is computed by integrating over the part of the surfac e that passes through the elements as well as through the thickness of the shell normal to the surface [3]. The shear load is applied by first computing the work equivalent nodal load for the nodes of the element that contains the edge on which the load is acting and then assembling this element load vector to the global external load vector.

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47 The tip deflection is shown in Figure 5 16 ( B ) with no scaling to show the magnitude of the deflection. The deflection is compared with the applied load in Figure 5 17 and listed in Table 5 4 Again the results from IBFEM and the exact solution reported in [ ref ] match very closely. Figure 5 15 Cantilever beam subjected to end shear load Figure 5 16 Cantilever beam model A). Mesh Model B) Cantilever plate deflection at P 2.4N. Figure 5 17 Deflection vs Load graph. B A

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48 Table 5 4 Vertical tip deflections for the cantilever loaded with end shear force. P W tip P W tip IBFEM Exact IBFEM Exact 0.2 0.651 0.663 1.4 3.982 3.912 0.4 1.291 1.309 1.6 4.350 4.292 0.6 1.957 1.992 1.8 4.710 4.631 0.8 2.570 2.493 2.0 5.042 4.933 1.0 3.071 3.015 2.2 5.389 5.202 1.2 3.496 3.488 2.4 5.701 5.444 Application in Flexural Hinge Design Flexural hinges and flexure based mechanism are used for many macro and microscale applications. Flexure mechanisms are designed to provide guided motion through elastic deformation. They can be used in place of sliding or rolling joints. They are general ly used in applications that demand high precision, design simplicity minimal assembly or long operating life [ 2 1 2 2 ]. In our example we use a single axis flexure hinge as shown in figure 5 1 8 [23] This flexural design has a circular cut out on either side of the blank which forms a necked down section. These kinds of flexures are popular for its simplistic design, ease of m anufacturing and high off axis stiffness. Figure 5 18 Dimensions, F orce s and coordinates of a flexural hinge

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49 The flexure used for this example can be defined with these physical and geometric parameters: , , A pressure of 100MPa is applied on the face that has the y coordinate as its normal. We model the hinge as a plane stress problem. The mesh used for the analysis using IBFEM is shown in Figure 5 1 9 and it consists of 9 Node BSpline elements. The mesh density for the model is with element size 0.25 A model for the same structure was also created in SolidWorks. For validation, the maximum displacement and Von mises stress contour at the neck computed in the structure using IBFEM and SolidWorks, is compared. Figure 5 19 FE Model Figure 5 20 Maximum Displacement comparison A). IBFEM B). SW A B

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50 Figure 5 21 Von mises stress contours A). IBFEM B). SW A B

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51 CHAPTER 6 CONCLUSION Summary The extension of implicit boundary method for geometrically nonlinear problems was described in this thesis and used to perform mesh independent analysis using a background mesh. Initially, IBFEM and its solution structure was explained. The governing equations for traditional finite element method were derived and explained. These equations were derived using Total lagrangian approach. Based on the IBFEM solution structure we deri ve the governing equations for boundary elements with essential boundary condition. The governing equation (4 8) derived using the Dirichlet function or the essential boundary function only affect the elements at the boundary with essential boundary condit ion imposed. For the rest of the elements the stiffness matrix computation and the residual load vector are calculated same as traditional FEM. The main challenge in the process is in accurately computing the stiffness associated with the fixed boundaries in the equation (4 14 ) (15) It was found that this stiffness is more accurately computed when it is assumed that within the transition region near the boundary the nonlinear part of the stiffness and the internal force are negligible. This is a reasonable assumption near fixed boundaries since the displacement near such boundaries will be small. There is a definite advantage in using the TL formulation for large deformation as opposed to the UL formulation when using a background mesh for analysis. The ele ments in the background mesh are undistorted regular shaped rectangles and cuboids which allow very accurate integration. In the TL formulation since the equations are written with respect to the original configuration the mesh in the original configuratio n remains undistorted and retains the advantage of

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52 accurate integration. Further research is needed on extending this approach for UL formulation which should be used only for materials where the constitutive relation can only be expressed with respect to Cauchy stress. The examples used for validation conclusively indicate the successful extension of implicit boundary method to geometric nonlinear problems. Scope of Future W ork Currently the computational time taken by IBFEM is greater than the computational time required by a traditional FEA software. This is because the shape functions for each element are recalculated for each iteration in a time step. Calculating a shape function at the start of a time step and using that for each iteration c an reduce the computation time. Also, the nonlinear solver used for these analysis uses a constant time step to move forward. Incorporating an arc length criteria or a similar criteria that helps fastens the process, could potentially reduce the computatio nal time. The examples discussed above use plane stress, plane strain, axis symmteric, 3D stress and 3D shell elements. Hence they do not cover all the elements. Element type like beam element and Mindlin plate element can also be validated.

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53 LIST OF REF ERENCES 1 Engineering, Transactions of ASME, Vol. 8., No. 3, 031005 (11 pages) doi:10.1115/1.2956990. 2 Bur uniform B Methods in Engineering, Vol 76, No. 13 pp. 1993 2028. 3 ent analysis of shell like 4 element analysis using non Numerical Methods in Engineering, Vol. 74., 1421 1447 5 Zhang Z. and Kumar A. V., Modal Analysis using implicit boundary finite element method, Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, DETC2014 35100, 2014 6 G.R. Liu, Mesh free methods: moving beyond the finite element method, CRC Press, Boca Raton, 2003 7 Y.T. Gu,. Meshfree methods and their comparisons, Int. J. Comput. Methods 2 (2005) 477 515. 8 P. Lanc aster, K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comput. 37 (1981) 141 158. 9 Y.Y. Lu, T. Belytschko, L. Gu, A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Engrg. 113 (1994) 397 414. 10 Z. D. Han, S. N. Atluri, The meshless local Petrov Galerkin (MLPG) approach for 3 dimensional elasto dynamics, CMC Comput. Mater. Con. 1 (2004) 129 140. 11 G.R. Liu, X.L. Chen, A mesh free method for static and free vibration analyses of thin plates o f complicated shape, J. of Sound Vibrat. 241 (2001) 839 855. 12 Y.T. Gu, G.R. Liu, A meshless local Petrov Galerkin (MLPG) method for free and forced vibration analyses for solids, Comput. Mech. 27 (2001) 188 198. 13 T. Belytschko, C. Parimi, N. Moes, N. Sukumar, S. Usui, Structured extended finite element methods for solids defined by implicit surfaces, Int. J. Numer. Methods Engrg. 56 (2003) 609 635.

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54 14 functions in Boundary value problems l. Mech. Rev., 48, pp. 151 188 1 5 Shapiro V, Tsukanov I. Meshfree simulation of deforming domains. Computer Aided Design 1999; 31:459 471. 1 6 H¨ollig K, Reif U, Wipper J. Weighted extended B spline approximation of Dirichlet problems. SIAM Journal on Num erical Analysis 2002; 39:442 462. 17 Klaus Jurgen Bathe and Arthur P. Cimento Some practical procedures for the solution of nonlinear finite element equations Computer methods in applied mechanics and engineering 22 (1980) 59 85. 18 Kim Nam Ho, Introd uction to Nonlinear Finite Element Analysis. Springer(2015) 19 K.Y. Sze, X.H. Liu, S.H. Lo, Popular benchmark problems for geometric nonlinear analysis of shells, Finite Elements in Analysis and Design 40 (2004) 1551 1569 20 H. Parisch, Large displacemen ts of shells including material nonlinearities, Computer methods in applied mechanics and engineering 27 (1981) 183 214. 21 Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York. 22 Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gordon & Breach, Amsterdam, Netherlands 23 Yingfei Wu and Zhaoying Zhou, 2002, Design calculations for flexure hinges, Review of Scientific Instruments 24 ComputerAided Geometric Design 2001; 18:195 220.

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55 BIOGRAPHICAL SKETCH Nikhil Bhosale was born in Pune, Maharashtra (India). He received his Bachelor of Technology in m echanical e ngineering from Visvesvaraya National Institute of Technology, Nagpur in 2012 After his graduation, he worked as a Senior Engineer at Bajaj Auto Ltd. in the Computer Aided Engineering department. He received the Employee achievement award in 2014 for his excellent initiatives during his work at Bajaj Auto Ltd In 2014 he enrolled in th e Master of Science program in m echanical e ngineering at Universi ty of Florida. He received his m echanical e ngineering in May 2016 from University of Florida. His a reas of specializati on include nonlinear f inite element analysis, background mesh finite element methods, numerical methods, optimization design and software development using Object Orientation Programming Techniques.