FEM : Finite Element Method 2017

Handout Instructor Hyukchun Noh ChungMoo 714 cpebach@Sejong.ac.kr Prerequisites Basic Statics and Mechanics Textbook Lecture Note

Handout References

Handout Syllabus

Handout Grading related

Handout NOTICE: Time is precious, and is related with the credit of a person To me, time must be a kind of ruler to grade a person So, please be on time at class If you do not present on time, even just a minute, you will be counted as absent: for Starting time & After short break No excuse for this  

Introduction to FEM(Finite Element Method) - Brief concept and Flow of FEM-

FEM ? - An approximate analysis scheme for natural phenomenon e.g. Solids, Electromagnetic field, Thermal effect and so on - FEM always gives WRONG answers Degree of accuracy is of importance - Do not believe FEM if you don’t know exact or analytical solutions Governing Equations Solution Domain Boundary conditions

Physical Phenomena are expressed by - PDE(partial differential equation) - Boundary conditions or Initial conditions Examples: Mechanical deformation of solids Transfer of thermal effects Flow of fluid Distribution of electric voltage ets. Boundary Value Problems (경계치 문제) FEM is just one of approximate Numerical schemes for solving above problems in nature

Properties Behavior Action: source FEM approximates the PDE in the form of Linear Algebraic Equations (선형대수방정식) instead of solving directly the PDE : Linear Algebraic Equation Properties Behavior (response) Action: source (Applied forces)

The properties, behavior and action are determined differently depending on the types of problem under consideration

The unknowns in the LAE(Linear Algebraic Equation) : Action, Behavior The unknowns are determined at nodes (discretized, pointwise) The nodes are located arbitrarily in the domain (depending on the analyzer) Nodes(절점) (Red dots) Red dots

(unknowns) Field of variable U interpolated by means of Shape Function and nodal values ui , in the figure below thus Continuous field is obtained Linear (1st order) Shape function Quadratic (2nd order) Shape function Nodal values Nodes Interpolated field using

Not-possible in general Representation by one equation ? Not-possible in general Divide the entire domain into sub-domains (i.e, finite elements) Example of Solid Element Elements (요소) Sharing the nodes between adjacent elements Shared nodes

The governing equation of an element is equal to that of entire domain Element equation: Single LAE, easily obtained by means of an approximation process Numerical integrations are used: e.g. Gauss-quadrature rule [Example] (Element equations)

After obtain all the element equations For from 1 to N number of FEs in the domain We can assemble them for the total domain leading to global equations By using Connectivity Global LAE

The global equation is solved by Nodal point displacement U: continuous (throughout the domain) Stresses can or cannot be continuous depending on element type Example of stress or displacement contour

Geometry (nodal coord.s) General Flow of FE Analysis Start Various schemes - Direct Inverse - Cholesky decomp. - Band - Skyline - Frontal - Iterative solvers and so on Form Load vector Read data Geometry (nodal coord.s) Connectivity Boundary condition Materials Analyze To obtain U Construction of Element Stiffness Determine Stresses U=De Assemblage to Global stiffness End Read data Actions

End of introduction to FEM