Numerical procedures_1
Page 1: Introduction to FEM
Overview of the course on Finite Element Method (FEM), Finite Difference Method, and Sampling Methods.
Page 2: Course Assembly
Assembled by Prof. Dr. Ir. Tom Molkens.
References include works by:
Klaus-Jürgen Bathe – Finite Element Procedures.
Andreas Öchsner & Markus Merkel – One-Dimensional Finite Elements.
Darrel Pepper & Juan Heinrich – The Finite Element Method: Basic Concept and Applications.
Includes lecture notes from various professors at KU Leuven.
Page 3: Table of Contents
Course topics:
Introduction and applications in civil engineering.
Formulation.
Shape function.
Matrix stiffness method applied on a tensile bar.
Local to global transformation.
Beam elements.
Equivalent loads.
Plane stress/strain elements.
Plate elements.
Practical reinforcement of slabs.
Finite difference method.
Sampling methods.
Page 4: Objective of Structural Engineering
Structural engineering art involves:
Molding materials not fully understood into shapes not fully analyzed.
Designed to withstand forces that are difficult to assess.
Page 5: Objective of the Course
Combine theory and exercises within 12 lessons.
Focus on:
Understanding basic principles of FEM.
Creating correct finite element models.
Assessing model restrictions.
Developing a critical attitude towards numerical results.
Page 6: Outline of Presentation
Topics covered:
Physical model and strong formulation.
Introduction to Finite Element Method.
Application examples.
Page 7: The Physical Model
Introduces the physical model relevant to FEM.
Key components:
Geometry of the domain.
Unknowns in the problem.
Field equations.
Boundary and loading conditions.
Initial conditions.
Page 8: The Field Equation
Field Equation: Often represented as differential equations.
Example equation structure for physical problems.
Page 9: Boundary Conditions
Complete problem description requires boundary conditions information.
Three main types:
Dirichlet: Fixed values.
Neumann: Fixed gradients.
Cauchy: Linear combination of both.
Page 10: The Strong Formulation
The strong formulation entails the set of differential equations and boundary conditions.
Solving it across the domain yields the exact solution.
Page 11: Example of Physical Model
Example: Circular bar of steel.
Geometry: Circular section.
External Load: Dead weight.
Boundary Conditions: Fixed at the top, free at the bottom.
Unknown Quantities: Extension (ΔL), strain (ε), and stress (σ).
Page 12: Field Equations in Example
Compatibility, constitutive law, and equilibrium are essential.
The equilibrium expression for an infinitesimal slice of the bar is described.
Page 13: Solving the Mathematical Problem
Steps to introduce boundary conditions.
Factors like displacements and equations to derive unknown constants.
Page 14: Solution of Example
Resulting displacement equation and expressions for strain and stress fields.
Displacement field's quadratic distribution is analyzed.
Page 15: Graphical Representation
Various graphical representations included to visualize the results and methodologies.
Page 16: Conclusion from Example
Strong formulation yields exact solutions in simple cases, but FEM is necessary for complex scenarios.
Page 17: Need for Computational Methods
Engineering problems' complexity often limits analytical solutions.
Need for efficient computational solutions in dealing with various boundary conditions and material behavior.
Page 18: Types of Computational Methods Required
Addressing complex geometries, loading cases, and non-linear material properties.
Page 19: Other Factors in Need for Computational Methods
Addressing imperfections and interactions with environmental factors.
Page 20: Experimental Needs in Research
Experimentation is cost-intensive; computational modeling aids in deriving design models with high accuracy.
Page 21: Conclusion
The importance of approximate solutions in FEM for differential equations of physical systems.
Need for controlled errors in solutions to maintain reliability.
Page 22: Historical Context of FEM
A brief historical overview from its inception at Boeing for aircraft design in 1953.
Expanded applications in civil engineering and other fields post-1960.
Page 23: Process of Finite Element Method
Key stages outline:
Discretizing domain into elements.
Establishing governing equations.
Assembling equations for the entire system.
Page 24: Finite Element Model Simplification
Highlights on how to streamline the model for computational efficiency.
Page 25: Types of Elements in FEM
Various element dimensions and types:
1D for bars, 2D for triangles and rectangles, 3D for bricks, etc.
Page 26: Conclusion on Applications
FEM is popularly applied in diverse engineering problems due to its versatility and practical application.