Detection Matlab Code For Finite

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This page contains links to MATLAB codes used to demonstrate the finitedifference and finite volume methods for solving PDEs. This page also containslinks to a series of tutorials for using MATLAB with the PDE codes.

The finite volume codes can handle non-uniform meshes and non-uniformmaterial properties. Therefore, these codes could be used or adapted topractical problems, and have been done so by me and others.

Here are direct links the web pages in this sereis. Short descriptions ofthe pages, with links, are given below.

Beware! I have tested the codes on a variety of demonstration problems.The codes are qualitatively correct for the test cases, and in several of thosecases I show that the code exhibit the correct asymptotic truncation error. Aswith any tool, however, these codes can be applied incorrectly or to asituation they were not designed to handle. Although the codes are designed tobe flexible, and have been tested fairly extensively, I do not guaranteethat they will be useful to you. Please let meknow if you find any bugs.

Finite-Difference Models of the Heat Equation

This page has links MATLAB code and documentation for finite-difference solutions theone-dimensional heat equation

where is the dependent variable, and are the spatial and time dimensions,respectively, and is the diffusion coefficient.

Finite Volume model of 1D convection

This page has links to MATLAB code and documentation for the finite volumemethod solution to the one-dimensional convection equation

where is the -direction velocity, is a convective passive scalar, is the diffusion coefficient for , and is the spatial coordinate.

Finite Volume model of 1D fully-developed pipe flow

This page has links to MATLAB code and documentation for the finite volume solution to the one-dimensional equation for fully-developed flow in a round pipe

where is the axial velocity, is the pressure, is the viscosity and is the radial coordinate.This is a simple and well-known flow with the exact solution

where is the pipe radius.

Finite Volume model in 2D Poisson Equation

This page has links to MATLAB code and documentation for the finite volumesolution to the two-dimensional Poisson equation

where is the scalar field variable, is a volumetric source term,and and are the Cartesian coordinates.This equation is a model of fully-developed flow in a rectangular duct, heat conductionin rectangle, and the pressure Poisson equation for finite volume models of fluid flow.

Set up MATLAB for working with the course codes

This page gives recommendations for setting up MATLAB to use the finite-differenceand finite-volume codes for the course. The main goals are to create a library folderfor storing the codes after downloading them, and setting up MATLAB so thatcode library is always included in the search path.

This page is part of a series of MATLAB tutorials for ME 448/548:

Set up MATLAB for working with the course codes – This page

Basic MATLAB Practice

This page walks you through some practice exercises to help develop your MATLAB skills.It's part of a series of MATLAB tutorials for ME 448/548:

Basic MATLAB Practice – This page

Practice with PDE codes in MATLAB

This page demonstrates some basic MATLAB features of the finite-difference codesfor the one-dimensional heat equation. This is a MATLAB tutorial without muchinterpretation of the PDE solution itself. Consult another web pagefor links to documentation on the finite-difference solution to the heat equation.

This page is part of a series of MATLAB tutorials for ME 448/548:

Practice with PDE codes in MATLAB – This page

Debugging MATLAB Code

This page walks you through some practice exercises to help develop your MATLAB skills.

Introduction to debugging

This page builds on the previous examples to introduce debugging. Any substantialprogramming effort involves a lot of debugging. Using the built-in debugger willmake that effort more productive.

In addition to the examples on the rest of the page, or maybe even before youwork through those examples, I recommend the following YouTube videos that introducethe debugger

MATLAB Debugging Tutorial (11 minutes) by Ilya Mikhelson
Lesson 4.5: Debugging (25 minutes) from the 2015 version of theCoursera course, Introduction to programming with MATLAB.

Both of those videos give a good introduction to the debugger. You probably only needto watch one. The second one provides a bigger perspectiveon MATLAB programming, and I got a lot out of it while watching at 1.5x speed. :-)

1D Spring elements finite element MATLAB code
This MATLAB code is for one-dimensional spring elements with one degree of freedom per node parallel to spring axis. This code plots the initial configuration and deformed configuration as well as the relative displacement of each element on them. Results are verified with examples of textbook; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
1D Beam elements finite element MATLAB code
This MATLAB code is for one-dimensional beam elements with two degrees of freedom per node (one translational -perpendicular to beam axis- and one rotational). Results are verified with examples of textbook ; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D Truss elements finite element MATLAB code
This MATLAB code is for two-dimensional truss elements (plane truss structures). This code plots the initial configuration and deformed configuration of the structure as well as the forces on each element. Results are verified with examples of textbook; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D Beam elements finite element MATLAB code
This MATLAB code is for two-dimensional beam elements (plane beam structures) with three degrees of freedom per node (two translational -parallel and perpendicular to beam axis- and one rotational); This code plots the initial configuration and deformed configuration of the structure. Results are verified with examples of textbook; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D Solid elements finite element MATLAB code
This MATLAB code is for two-dimensional elastic solid elements; 3-noded, 4-noded, 6-noded and 8-noded elements are included. This code plots deformed configuration with stress field as contours on it for each increment so that you can have animated deformation. Results are verified with Abaqus results; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D Penalty and Lagrange Multiplier Contact finite element MATLAB code
This MATLAB code is for two-dimensional elastic solid elements with contact algorithms implemented in it. Penalty method and Lagrange multipliers contact algorithms are implemented. This code is also using large deformation formulations. Results can be animated.; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D Large deformations finite element MATLAB code
This MATLAB code is for two-dimensional elastic solid elements with large deformations (Geometric nonlinearity). Both Updated Lagrangian Method and Total Lagrangian Method are available. I have used Newton-Raphson solver. Green-Lagrange strains are used in these codes. The results can be animated.; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
2D XFEM for Crack eXtended finite element MATLAB code
Mohr-Coulomb plasticity tangential stick-slip rule, penalty method, and Elastic predictor-Plastic corrector algorithm were used. Results are verified with Fushen Liu and Ronaldo I. Borja, “A contact algorithm for frictional crack propagation with the extended finite element method”, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
3D Truss elements finite element MATLAB code
This MATLAB code is for three-dimensional truss elements (space truss structures). This code plots the initial configuration and deformed configuration of the structure as well as the forces on each element. Results are verified with examples of textbook; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
3D Solid elements finite element MATLAB code
This MATLAB code is for three-dimensional elastic 8-noded solid elements; This code plots deformed configuration with stress field as contours on it for each increment so that you can have animated deformation. Results are verified with Abaqus results; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.
3D Large deformations finite element MATLAB code
This MATLAB code is for three-dimensional elastic solid elements with large deformations. Both Updated Lagrangian Method and Total Lagrangian Method are available. The results can be animated. ; arbitrary input geometry, nodal loads, and material properties for each element can be defined by user.

What is Finite Element Method?

The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis (FEA). Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. Read more about Finite Element Methods

What is MATLAB?

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment. A proprietary programming language developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, C#, Java, Fortran and Python.

What reference books did I use for my codes?

Some of the main textbooks I used for codes are:
General concepts and basic codes:

Finite Element Procedures by K.J. Bathe
The Finite Element Method (v.1 & v.2 & v.3) by Zienkiewicz & Taylor
Fundamental finite element analysis and applications by M. Asghar Bhatti

Contact mechanics:

Computational Contact Mechanics by Peter Wriggers