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Lecture# Finite Element Method: Local Approach

Description

This lecture covers the formulation of finite element method with a focus on the local approach, including the elementary matrices and vectors of rigidity and external forces, assembly operations, and examples of finite element assembly and application. It also discusses the stiffness matrix, change of landmark, global stiffness matrix, system of linear equations, and practical application examples using MATLAB.

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Related concepts (34)

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Direct stiffness method

As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM).

Elementary matrix

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form.

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

ME-372: Finite element method

L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à applique

Related lectures (13)

Finite Element Method: Local ApproachME-372: Finite element method

Explores the local approach of the finite element method, covering nodal shape functions, solution restrictions, sizes, boundary conditions, and assembly operations.

Finite Element Method: Global vs Local ApproachME-372: Finite element method

Compares the global and local approaches of the Finite Element Method.

Finite Element Method: Local ApproachME-372: Finite element method

Explores the local approach in the finite element method, covering nodal shape functions and assembly.

Finite Element Method: Higher Order ModelsME-372: Finite element method

Explores precision of higher order finite element models and applications of quadratic finite elements in elastodynamics.

Finite Element Method: Weak Formulation and Galerkin MethodME-372: Finite element method

Explores the weak formulation and Galerkin method in Finite Element Method applications, including boundary conditions and linear systems of equations.