ENG I工一 212 TnT5T6
This lecture aims to introduce advanced finite element methods (FEMs) for linear and nonlinear solid mechanics. Topics like FEMs for fluid flow problem and meshfree methods may also be included depending on the overall progress of the lecture. Computational tools in solving nonlinear problem will also be demonstrated in the lecture.
Course keywords: Finite Elements Methods, Nonlinear Solid Mechanics, Variational Equations, Finite Strain Theory Course Description: To introduce advanced finite element methods (FEMs) for linear and nonlinear solid, mechanics, multi-field variational principles, total Lagrangian and updated Lagrangian formulations, nonlinear material models and numerical procedures for path independent (hyperelasticity) and path dependent (plasticity) problems. Topics like fluid flow problem and meshfree methods may also be included depending on the overall progress of the lecture. Prerequisites: Finite Elements Methods Grading: Homework 40%; Computer Assignment 40%; Final Project 20% Lecture Material: In-class notes, and lecture handouts will be given. There is no required textbook for this course. However, textbooks in the reference are recommended. The FEniCS with Google colab will be used as the simulation platform. Reference: T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover, 2000. Belytschko, T., Liu, W. K., and Moran, B., and Khalil Elkhodary, "Nonlinear Finite Elements for Continua and Structures”, 2nd Edition, John Wiley & Sons, 2014. Finite Element in Computer Science (FEniCS): https://fenicsproject.org/ Course Outline: 0. Fundamental Knowledge Strong form and Weak Form Galerkin Equation Basics of Finite Element Formulation Introduction to FEniCS, Google colab and Paraview 1. Constrained Problems Numerical Difficulties in Nearly Incompressible and Incompressible Elasticity Volumetric Locking and Remedies Reduced Integration with Hourglass Stabilization, Selective Reduced Integration Reissner-Mindlin Plates and Shear Locking Shear Locking and Remedies 2. Multi-Field Variational Principles Hu-Washizu Variational Principle Hellinger-Reissner Variational principle Complementary Energy Principle Hybrid Stress Formulation Mixed (up) formulation for Incompressible Problems 3. Large Deformation Problems in Solid Mechanics Nonlinear solver: Newton Raphson methods Total and updated Lagrangian formulation History independent materials: hyperelasticity History dependent materials: plasticity Linearized buckling problems (Depending on progress) Contact problems (Depending on progress) 4. Finite Element Methods for Fluid Problems (Optional Topics) Advection-diffusion (AD) problem and its numerical difficulties Petrov-Galerkin formulation for AD problems Navier-Stokes equations Streamline upwind Petrov–Galerkin (SUPG) formulation Variational Multiscale Stabilization (VMS) formulation Aribitrary Lagrangian Eulerian method (ALE) for Fluid-Structure Interaction problem 5. Meshfree methods (Optional Topics) Moving Least-Squares Approximation Reproducing Kernel Approximation Galerkin Meshfree Methods for Solving PDEs
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