NONLINEAR PROGRAMMING SOLVERS FOR HYBRID FINITE ELEMENT

Open Access
Author:
Thukral, Payal
Graduate Program:
Civil Engineering
Degree:
Master of Science
Document Type:
Master Thesis
Date of Defense:
November 28, 2017
Committee Members:
  • Dr. Kostas Papakonstantinou, Thesis Advisor
  • Dr. Gordon Warn, Committee Member
  • Dr. Ali Memari, Committee Member
Keywords:
  • Nonlinear finite element analysis
  • Newton's method
  • Nonlinear programming
  • Sequential quadratic programming
  • Crisfield's arc length method
  • Conjugate gradient method
  • Path continuation
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.