What is finite element: Definition and 115 Discussions
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points.
The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.
The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations.
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
I have been watching Mike Foster's video series of the Finite Element Method for Differential Equations (FEM). In this episode, he solves a DE relating to temperature. As the final step, he gives the following equation: $$[K] [T] = [F]$$ In this equation, I understand that ##[K]## is the...
My goal is to numerically solve (finite elements using FEniCSx software) Ohm's law ##\vec J = \sigma \vec E##, where ##\vec E = \nabla V##, ##\vec J## is given (the current density is given on some boundaries), and ##\sigma## is algo given (the electrical conductivity). The problem is solved...
I know that if there is only one conductor providing the current density, then the current density can be used.
But if you apply Maxwell's equation when there are multiple current sources, I don't know which value to use.
This is not an analysis using a tool, but a problem when I develop the...
I know how to apply boundary condition like Dirichlet, Neumann and Robin but i have been struggling to apply divergence free condition for Maxwells or Stokes equations in nodal finite element method. to overcome this difficulties a special element was developed called as edge element but i don't...
the simple rectangular isoparametric element (curved edges element) can be used to generate many complex shapes like..
square to circle
square to triangle
two square elements to annular.
and the results i calculated in python code (2D case) are very accurate, then i confused why to use complex...
I recall, about 30 years ago, seeing the eight mode shapes (and eigenvalues) of a single plane stress (or strain) finite element.
Also, years ago, I wrote a FORTRAN code to obtain them (using IMSL libraries)
I know there are eight (because the plane quad element had 8 degrees of freedom)
I...
I want to build a high performance computer optimized for FEA. If anyone has any suggestions or experience with this, please chime in.
I'm looking for suggestions on processors, RAM, video card, ALL of it!
Is there any existing work focusing on analyzing finite element visualized results and how they differ from analytical solutions, rather than on the method?
I would like any tips about a Maple ''home made'' program that I received for a project but this program seems to stop before the very end of the code. I want to find de lift of an airfoil with Boundary finites elements method. I have this error at the very end :
Error, (in fprintf) number...
Hello engineer's community !
I got a problem. If i got a truss frame with 2 bars and an initial nodal displacement for the common node , then how can i calculate the velocity of the oscillation as function of time ? (the 2 bars have different length , same elasticity modulus , same density and...
Clearly e ∈ N. If a, b ∈ N, say ##a^k = b^l = e##, for some k,l ∈ N, then ##(ab)^{kl} = (a^k )^l (b^l )^k = e^l e^k = e##; thus, ab ∈ N. Also, ##a=a^{−1}##, so ##a^{−1}## ∈ N. Thus, N is a subgroup. As G is abelian, it is normal. Take any c ∈ G. If, for some n ∈ N, we have ##(cN)^n = eN##...
In general, one could say the Finite Element Method is merely an interpolation method that could be used to solve field equations. Despite that, this question focuses exclusively on the FE Method and its use in Mechanical Engineering.

I have noticed that some schools now...
Unlike in electromagnetics, the nonlinearity of mechanical structures is not only due to the material property. It could be due to large deformation and contact as well. Even though I will be implementing the existing and popular methods, I am still scared and I feel it is not a job that can be...
My background is BS Math with Physics minor (3.5) and MA Math (3.7) from the University of Houston. My current plan is to explore the necessary background for doing research in finite element analysis. Over the next few semesters, I plan to take Numerical Methods for PDEs, Statistical Computing...
In the formulation of EulerBernoulli Beam Theory, there are two degrees of freedom at a point, w and dw/dx. Typically, the finite element model of this theory uses cubic polynomial for interpolation of $w$ using a two noded element as given in Chapter 5 of this book [1]. This element is a...
Hi!
I have a code that solve the poisson equation for FEM in temperature problems.
I tested the code for temperature problems and it works!
Now i have to solve an Electrostatic problem.
There is the mesh of my problem (img 01).
At the left side of the mesh we have V=0 (potencial).
There is a...
Does anyone know of a great text that shows how to set up FEA in MATLAB in detail. I have wrote out the sort of pre pseudo code from John Andersons  Computational Fluid Dynamics now I just need to build it out in code. I am old school with the hand sketches, hopefully this does not get me a...
Hello,
Does anyone know if a finite element skeleton source code exists. Finite element can be used for structural memebers, gas dynamics, etc. But the over all stepping software is the same and then just input the core equations.
My application is for pseudo one dimensional compressible...
Homework Statement
I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
Hi, I would like to ask why center of element displacement is always estimated with polynomial equation involving nodes displacement (like in attachment/picture)? Also I know that if nodes' number increase for an element then displacement of center of element is estimated with higher order...
Homework Statement
Linear and quadratic elements differ because of the extra midnodes on quadratic elements. Quadratic elements can "bend", linear can't. How do you define the DOF of a element? (see at solution attempt)Homework Equations

The Attempt at a Solution
For example: a linear...
HI, initially I would like to put into words that there are sufficient resources, books or lecture videos on YouTube related to finite element analysis especially for structural dynamics (for instance JURGEN BATHE in MIT). But I would like to make you sure that there NO lecture videos or other...
Hi. Can someone point me to a good (and really basic) PDF file or text regarding Finite Element analysis? I would prefer it having an example where it would solve the 1D heat equation or the Laplace equation so I can extend what I know from finite difference methods to it.
Sorry if this seems...
Homework Statement
In the third picture , I don't understand the circled part , add up the values in the diagonal .. How to do that ?
I don't understand how to get k13 , k14 , k21 , k22 , k33, k41 and k42 .
Homework EquationsThe Attempt at a Solution
As we see in the second picture , the k21...
Okay, I'm following a series of video lectures on applications of finite element method to engineering, and the tutor started by demonstrating the mathematical background of FEM using a simple heat transfer problem. He derived the governing equation (in just one dimension):
(1)...
I am solving some 2nd order differential equations using the finite element method. Doing so I represent the second order derivative at a given point as:
∂2ψi/∂x2 = 1/(Δx)2 (ψi1+ψi+12ψi)
And solve the differential equation by setting up a matrix of N entries and solving for the eigenvectors...
I would like to know how to implement internal hinges in a program I'm developing. A hinge is created by changing the stiffness matrix of the beam. The problem is when two interconnected beams have a hinge at the same location, so basically we have a hinged joint, in this scenario I will obtain...
I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phiindependence on the potential, therefore the problem is essentially 2D; I am working only with...
Hi,
I am running a finite element on a cylinder with that converges at the bottom for a opening, which is symmetrical in both directions so i modeled one quarter but the problem is my stresses are the same with when i compare with a full model that i also done but the deflections are different...
Homework Statement
The potential energy of an element in terms of local displacements q1 and q2 is given by the expression:
Π = 3(q1^2)  6q1q2 + 9(q2^2) + 9q1
Write down the expressions for element stiffness matrix k and element force f.
I am at a complete loss as to what...
I was reading the finite element method in engineering by Rao and in the first example he ends up with a matrix that is singular.
The matrix is the following:
\begin{pmatrix}
2 &2 & 0\\
2 & 3&1\\
0&1& 1
\end{pmatrix}
Which is a symmetric matrix as far as I can remember...
Hi guys, :smile:
I am a mechanical engineer, and want to learn finite element analysis.
I want to know what is the best book to start with.
Assume I have no prior knowledge of the subject. :redface:
Thanks,
Sety.
What are the advantages and disadvantages of both AEM and FEM and which on is easier.
I am doing a project and I should use one of these two methods to solve for a truss system.
P.S. computer programming shall be used.
In the end which method is better for this case?
A simple question about elasticity theory/finite element method:
Suppose I have a tetragonal 2D piece of a linear isotropic elastic material, that has Young's modulus ##E## and Poisson's ratio ##\nu##. The vertices of the tetragon are at positions ##\textbf{x}_{1}##, ##\textbf{x}_{2}##...
Hey! :o
Given the following twopoint problem:
$$y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$
$$y(0)=0, y'(1)=my(1)$$
where $ b \in C^1([0,1];R), f \in C([0,1];R)$ and $ m \in R$ a constant.
Give a finite element method for the construction of the approximation of the solution $y$ of the...
Hi,
I have been stuck on a problem for a while now (3.24 part c).
My attempt is as follows:
Internal virtual work = external virtual work
T/2 ∫0>L (∂u/dx)(∂v/dx)dx + ∫0>L (∂^2u/∂t^2)vdx = ∫0>L (Pv)dx
Stationarity is already invoked on this functional as it's the principle of...
I developed finite element program (MFEM) in java for BVP &IVP to compute partial differential equation.
I am facing one problem and description is as follows
my problem is on generalized eigenvalue problem generated in wave propagation through rectangular wave guide in TE mode. (Differential...
I am thinking about taking a finite element method course. I know what FEM is and how it solves boundary value problems and stuff but I'm wondering how widespread it is used...
Is it a useful numerical technique? What industries/research use it? I am interested in research in continuum...
Hi,
This is a sample problem from logan finite element method. I have attached the problem and solution given in the book. As per the problem i first derived the stiffness matrix and den putting the boundary conditions started solving for the forces. I am stuck as three forces are unknown but...
Dear all,
Please tell me the steps are used to find Modal loss factor of composite constrained layer damped beam by using Finite Element software such as ANSYS etc.
Hello
my question is basic but i would like to know this and get explanation
I a have simple question, about meshing,I know from finite element the it is always better to have node on node of the mesh for two different parts.But if i have node of mesh of on part intersect with line of...
Hello,
Please help my question is really basic,but i need help
I have concrete box and steel box and bolt as illustrated in picture one(quarter shape),so my questions are:
the key points in symmetry for the quarter of the shape are:
1.prevent translation in the two plan...
Homework Statement
Consider the problem
$$u''\left(x\right) = 1, \;\; 0 < x < 3, \;\; u \left(0\right) = 0, \; u' \left(3\right) = u\left(3\right)+1.$$
Formulate a MATLAB code to produce the solution and plot the solution from 0 to 3.
Homework Equations
The Attempt at a Solution
Multiply by...
Howdy,
I am trying to formulate a proof to show that the shape function
[N(x,y)] = [Ni(x,y) Nj(x,y) Nk(x,y)]
and the the basis functions
Ni(x,y) = (1/2A)(ai + bix + ciz)
Nj(x,y) = (1/2A)(aj + bjx + cjy)
Nk(x,y) = (1/2A)(ak + bkx + cky)
are valid for triangular, 2 dimensional...
Homework Statement
Hey guys, my problem is not really the end solution, but the preliminary step that I'm not too sure about. Given the following diagram:
The image on the left is the given model, and the right image is my nodal element FBD, but somehow i don't believe that's correct. Can...
Anyone know of some good, free finite element analysis software?
I used NASTRAN in college, but from what I can tell that costs at least several thousand.