# NONLINEAR PROGRAMMING SOLVERS FOR HYBRID FINITE ELEMENT

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NONLINEAR PROGRAMMING SOLVERS FOR HYBRID FINITE ELEMENT

ABSTRACT

The focus of this work is on analyzing and developing nonlinear solvers for performing nonlinear

structural analysis for large displacements in both elastic and inelastic cases. The response of a structure to a load application is shown by its equilibrium path which may include snap back and snap through behavior. Material and geometric nonlinearities are taken into consideration while developing the response path of a multi-element structure with sections discretized using fiber elements. Traditionally, Newton’s method is employed for solving the system of nonlinear equations but it comes with certain challenges. The response determination becomes difficult when stiffness matrix becomes singular at turning point. It also requires the calculation of the inverse of a Hessian matrix, which is costly. Newton’s method gives quadratic convergence but as the scale of the structure increases, resorting to Newton’s method becomes difficult.

These limitations motivate us to explore new solvers. Hence, in this study we analyze and develop various nonlinearly constrained optimization solvers for a recently suggested hybrid finite element. In particular, we compare the performance of conjugate gradient method with or without preconditioning, Sequential Quadratic Programming method and augmented Lagrangian method. For the case of structural response with snap back and snap through behavior, a new method called the implicit path continuation method is developed to ensure path continuation and solution convergence. The various solvers are then validated by obtaining responses of three benchmark structural problems with large displacements and rotations, and comparing the results with the conventional Newton’s method and a variant of Newton’s method with submatrices.

List of Figures………………………………………………………………………………………………………………….. vi

List of Tables…………………………………………………………………………………………………………………. viii

Acknowledgements…………………………………………………………………………………………………………… ix

Chapter 1 INTRODUCTION………………………………………………………………………………………………… 1

1.1 General…………………………………………………………………………………………………………………. 1

1.2 Research……………………………………………………………………………………………………………….. 3

Chapter 2 LITERATURE REVIEW……………………………………………………………………………………….. 5

2.1 Finite Element Analysis……………………………………………………………………………………………. 5

2.1.1 Displacement based finite element method…………………………………………………………… 6

2.1.2 Force based finite element method…………………………………………………………………….. 7

2.2 Nonlinear Programming…………………………………………………………………………………………… 8

2.2.1 One Dimensional Unconstrained Optimization……………………………………………………. 11

2.2.2 Unconstrained optimization……………………………………………………………………………. 13

2.2.3 Constrained optimization……………………………………………………………………………….. 25

Chapter 3 PROBLEM STATEMENT…………………………………………………………………………………… 31

3.1 Finite Element problem………………………………………………………………………………………….. 31

3.2 Method 1- Newton’s method……………………………………………………………………………………. 37

3.3 Method 2-Newton’s method with submatrices……………………………………………………………… 39

3.4 Method 3-Conjugate gradient method with and without Preconditioning……………………………. 43

3.5 Method 4- Sequential quadratic programming method…………………………………………………… 44

3.5.1 Null space calculation……………………………………………………………………………………. 46

3.5.2 Preconditoning of matrix……………………………………………………………………………….. 46

3.5.3 Algorithm to solve Problem 2…………………………………………………………………………. 47

3.5.4 Hybrid Finite element problem………………………………………………………………………… 48

3.7 Method 5-Augmented Lagrangian method………………………………………………………………….. 49

3.8 Iterative scheme for traversing snap back and snap through…………………………………………….. 50

3.8.1 Method 6-Arc Length method with Newton’s iteration…………………………………………. 50

3.8.1.1 Rik’s method [17]…………………………………………………………………………………………….. 54

3.8.1.2 Crisfield’s method [15]……………………………………………………………………………………… 54

3.8.2 Method 7- Implicit path continuation: Modified Homotopy method……………………………….. 56

3.8.2.1 Homotopy-Motivation for this method………………………………………………………………………. 56

3.8.2.2 Implicit path continuation method for post critical behavior determination for hybrid finite element analysis…………………………………………………………………………………………………………….. 58

Chapter 4 RESULTS………………………………………………………………………………………………………… 63

4.1 Cantilever with vertical load at free end……………………………………………………………………… 63

4.2 Cantilever with moment at free end…………………………………………………………………………… 66

4.3 Toggle frame……………………………………………………………………………………………………….. 67

4.4 Lee’s Frame…………………………………………………………………………………………………………. 70

Chapter 5 CONCLUSIONS………………………………………………………………………………………………… 72

# Chapter 1 INTRODUCTION

## 1.1 General

In structural analysis, there is an increasing demand for performing analysis of large scale structural systems with nonlinear behavior. The load response of a structure can be seen physically in the form of deflections, rotations or vibrations. For a preliminary analysis, the material behavior is considered linearly elastic, i.e. stress (axial or bending) varies linearly with the corresponding strain and upon unloading the body recovers back to its original configuration. This is an idealized model which is very straightforward for obtaining structural element behavior. The stiffness matrix is constant in this case. The structural deformations can be also superimposed for this linear case.

However, for real life problems with large loads many complexities arise due to nonlinearities. Nonlinear phenomena arise due to physical variables related by nonlinear quantities. In nonlinear structural analysis, the relationship between the stress and strain becomes nonlinear and the stiffness matrix does not remain constant. When a structure is tested for large loads up to collapse, we observe bifurcations and instabilities in structural response. Hence, we define two types of nonlinearities: material and geometric. Material nonlinearly is commonly seen in all elements. As the name suggests, this type of nonlinearity is attributed to material behavior, as the force increases the stress-strain behavior becomes nonlinear. In many structures, geometric nonlinearity is also observed due to large displacements and large rotations. Geometric nonlinearity arises when the change in geometry becomes significant and is taken into account in equilibrium and compatibility equations. For small deformations, geometric nonlinearity does not come into picture as it introduces very small error. But as the displacements and rotations increase in magnitude,

Figure 1 Material Nonlinearity

Figure 2 Geometric Nonlinearity

the frame of reference is significantly different from the original one and the stress-strain curve of body is nonlinear.

To completely define the behavior of continuum problems, it is required to have deformation information for all the sections of the body. This is a very cumbersome task since initially we know only about the applied loading and the general body configuration. To obtain structural deformations and internal forces for a body, we need a solution strategy so that we scan define, analyze and obtain structural responses. Hence finite element methods are used which are very popular in several engineering fields and provide approximation to boundary value problems.

In displacement based finite element methods, compatibility is strictly enforced due to their formulation, but equilibrium of forces is not exact which results in certain errors. In force-based finite element methods, the nodal forces are unknowns and exact values of forces are obtained. Since compatibility is implemented by integrating, the deformations have now some errors in this case. In order to eliminate the limitations of these methods, mixed and hybrid finite element methods have been proposed in several works. From the available studies in the literature there is still need for development of methods that can overcome several limitations of current techniques.

## 1.2 Research

This work is based on developing computational solutions which enhance the efficiency of hybrid finite element methods. In this regard, we explore various optimization techniques which are being successfully implemented in other fields of engineering. The aim is to explore optimization algorithms for solving the finite element problems by improving upon the classical methods. We want to improve the accuracy of solution and computational cost. In this regard, a hybrid finite element model was developed by [1] for large displacements and rotations and Newton’s method has been used to iteratively obtain the structural response. Hence the overall goal of this work is to search for alternative methods than second order Newton’s methods in solving problems with multiple degrees of freedom based on this recently suggested hybrid element.