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).

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  1. Vanilla Gorilla

    B Incorporating boundary conditions in the Finite Element Method (FEM)

    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...
  2. fluidistic

    I Finding the weak form of an eq. with DG elements (finite elements)

    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...
  3. JH_1870

    A A question about current density in finite element analysis

    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...
  4. M

    How to apply divergence free (∇.v=0) in nodal finite element method?

    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...
  5. M

    A When should you use more mesh elements in the finite element method?

    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...
  6. T

    The 8 Mode Shapes of a plane finite element

    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...
  7. F

    What's the best shape function/element in finite element methods for phase field simulation?

    What's the best shape function/element in finite element methods for phase field simulation?
  8. KyleGranger

    Best high performance computer build for finite element simulations

    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!
  9. F

    Existing work on finite element result analysis

    Is there any existing work focusing on analyzing finite element visualized results and how they differ from analytical solutions, rather than on the method?
  10. C

    Maple Boundary finite element method in Maple

    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...
  11. P

    Engineering Finite Element Analysis / Structural Analysis

    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...
  12. R

    Show N is a normal subgroup and G/N has finite element

    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##...
  13. T

    Abuse of the Finite Element Method

    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...
  14. H

    Nonlinear Structural Analysis with Finite Element Method

    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...
  15. Shackleford

    PhD in Applied Mathematics - Finite Element Analysis

    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...
  16. A

    Finite Element Model of Euler-Bernoulli Beam Theory

    In the formulation of Euler-Bernoulli 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...
  17. Alan Lins Alves

    Problem with the Finite Element Method applied to Electrostatics

    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...
  18. R

    MATLAB Finite Element Analysis (FEA) in MATLAB - Textbook Guide

    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...
  19. R

    MATLAB Finite element skeleton code for matlab

    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...
  20. C

    Finite element solving of Laplace's equation doesn't converge

    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...
  21. mertcan

    Finite element analysis node displacement

    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...
  22. E

    Finite Element Method: Linear vs. Quadratic Elements & DOF

    Homework Statement Linear and quadratic elements differ because of the extra mid-nodes 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...
  23. mertcan

    Studying Finite element analysis for Electromagnetism

    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...
  24. maistral

    Discussions about Finite Element Analysis (PDFs)

    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...
  25. F

    Understanding Finite Element Method: Diagonal Values and Calculating K Matrix

    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...
  26. ramzerimar

    I Finite Element Method: Weak form to Algebraic Equations?

    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)...
  27. S

    Finite Element Method: Find Total Mass Matrix of Coupled Element

    Does anyone know how to find the total mass matrix of two coupled element by using Finite Element Method?
  28. A

    Nonuniform finite element method

    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 (ψi-1+ψi+1-2ψi) And solve the differential equation by setting up a matrix of N entries and solving for the eigenvectors...
  29. M

    Finite element method for frames/beams

    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...
  30. C

    Can You Help With Finite Element Analysis in Cylindrical Coordinates?

    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 phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
  31. S

    Finite Element Symmetry Problem

    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...
  32. F

    Finite Element Analysis: PE of an element

    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...
  33. Chacabucogod

    Solving the Finite Element Method Matrix with Rao - Engineering

    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...
  34. S

    What is the best book to start learning Finite Element Analysis?

    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.
  35. F

    Finite element method and applied element method

    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?
  36. hilbert2

    Eq. of motion of elastic 2D finite element

    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}##...
  37. M

    MHB Finite element method for the construction of the approximation of the solution

    Hey! :o Given the following two-point 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...
  38. C

    Finite element procedures book - Bathe

    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...
  39. M

    Help with finite element program I developed for Diff Eqns

    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...
  40. C

    Is a Finite Element Method Course Right for Me?

    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...
  41. A

    Solve Grad Shafranov Equation using Finite element method?

    I want to compute the flux surfaces using FEM but i haven't found any good source to read. any help will be appreciated. Thank you
  42. C

    Problem in finite element method using direct stiffness method

    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...
  43. H

    Determination of Modal loss factor by Finite Element Software

    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.
  44. S

    Finite Element Meshing: Understanding Node Intersections

    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...
  45. S

    Symmetry in finite element

    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...
  46. S

    Finite Element MATLAB Code for Solving Boundary Value Problems

    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...
  47. T

    Valid Basis Functions for Triangular 2D Elements: Solving for Coefficients

    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...
  48. F

    Finite Element analysis: Free body diagram help

    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...
  49. A

    Free Finite Element Analysis Software

    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.