What is Finite element method: Definition and 53 Discussions

The finite element method (FEM) is a widely used 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 uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).

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  1. M

    Finite element method approximation

    In the picture above, we have on the left a figure representing spatial element, and on the right a figure representing reference element. The type of element used in the method here is a linear quadrilateral element. What I understand is moving or mapping from the reference element domain...
  2. Vanilla Gorilla

    B Solving PDEs via FEM: A Gut Check of the Procedure

    Hi, I am a newcomer to FEA/FEM. I am aware that for any practical purpose, software is used to solve problems. However, before I utilize software, I want to understand the process the computer goes about doing. This document is simply an attempt to summarize the solving procedure in a concise...
  3. 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...
  4. T

    A F.E.M and Hamilton's Principle (converting differential equations into integral equations)

    Hello May I begin by saying I do not exactly know what I am asking, but here goes... In the Finite Element Method (as used in Solid Mechanics), we convert the differential equations of continuum mechanics into integral form. Here, I am thinking of the more pragmatic Principle of Virtual Work...
  5. M

    A Rectangular higher order edge element (finite element method)

    I have solved many finite element problems using nodal based (rectangular element) for higher order. now i am trying to solve electromagnetic problem using vector element (Nedelec or Whitney). I know only triangular edge based element with first order only and not higher order. i am searching...
  6. Mankoo

    What is the one-dimensional counterpart to the Green-Gauss theorem?

    Are my answers to a and b correct? a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer:The derivative b) In a three-dimensional situation, a volume integral of a divergence of a vector field can...
  7. M

    A Calculate the work done by pressure rupturing a spherical containment

    I am post-processing a simulation. A spherical is meshed by many little triangles. A time-dependent pressure (p=10*t) is equally applied to the inner surface of a spherical in the normal direction all the time. After t1=0.1s, the spherical is broken, and each little triangle is disconnected...
  8. 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...
  9. 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...
  10. M

    Treatment of source term on the boundary: FEA and Comsol

    I would like to solve a coupled system of two PDEs using Comsol for the following geometry: Equation 1 (valid for 0⩽Z⩽bm): The initial and boundary conditions are: Tm(r,t→0)=20 Tm(r→rw,t)=70 Tm(r→∞,t)=20 However, for bm⩽Z⩽bm+b2, the equation to solve is: With the following initial and...
  11. I

    What size is the Global Stiffness Matrix in this Example?

    does this Beam, composed of three elements and 4 nodes(considering lateral deflections and slopes) has an 8x8 global stifness matrix and if so is the global matrix calculated the same way as a 6x6 stifness matrix for the same kind of beam but only with two elements and 3 nodes
  12. 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...
  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. A

    "Assembly" of elements of different dimensionalities (1-D, 2-D)

    I am currently working on a problem which involves both 1D and 2D elements. I know that these elements have different degrees of freedom at a node and, therefore, the procedure of assembly of elements will different from a purely 1D or a purely 2D mesh. I don't know the rest of the details. Any...
  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. H

    Total energy (ETOTAL) is not zero and increasing (Abaqus)?

    Hi all, I have performed a small static analysis with one element (C3D8). The bottom face of element is completely constrained, and the top part is displaced. In the geometrical linear analysis, the internal energy (ALLIE) and the work done (ALLWK) are equal. So, with ETOTAL = ALLIE - ALLWK...
  19. S

    Why are 3 planes needed to define stress at a point?

    My question is simple. Why do we need 9 different quantities, ie 1 normal stress and 2 shear stresses on 3 different planes, to define stress at a point? example: http://www.geosci.usyd.edu.au/users/prey/Teaching/Geol-3101/Strain/stress.html I think it should be enough to define the 3 stresses...
  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. 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...
  22. 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...
  23. lawa44

    Direct Stiffness Method for a distributed load

    I'm working on a question which asks to determine the deflection, curvature, forces and moments of a simply supported beam with a distributed load. diagram shown in here http://imgur.com/a/wpI4kInitially, I've done the calculation with 1 plane beam element with 2 nodes. At L = 0 and L = 3. But...
  24. 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)...
  25. 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?
  26. Emre Yucel

    Which US Graduate Schools are Best for Mechanical Engineering Research?

    Hello, My name is Emre and I am a MSc student in mechanical engineering.I am looking for PhD in US.My research interests are mechanical vibrations,rail vehicles,finite element method and control theory.In fact I have 2.68/4 GPA in Undergrad and 3.79/4 GPA in Master.So if anyone has any advice...
  27. ramzerimar

    Software for learning structural analysis and FEM?

    I'm in the third semester of mechanical engineering and I'm taking classes on engineering statics right now. Our professor said the he will give us a preview of finite element methods at the end of the semester. Which softwares would you recommend for learning basic structural analysis? I mean...
  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. L

    Finite-element solution of Sturm-Liouville problem

    Can someone point me to a simple introduction to the use of finite-element methods to find eigenvalues and eigenfunctions for a Sturm-Liouville problem? In my searching I've seen various suggestions that this is now the dominant method, but I've had no luck in finding an actual explanation, with...
  31. 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...
  32. 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?
  33. 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...
  34. 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...
  35. 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
  36. 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...
  37. 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...
  38. G

    Solving Neutron Problems with Commercial Finite Element Method Codes

    As I know, the method to solve neutron problem is divided into two steps now, neutron transport calculation for fuel assemblies and neutron diffusion calculation for whole reactor core, both using specified code such as CASMO and SIMULATE from STUSVIK. I want to know whether the commercial...
  39. H

    Dynamic analysis using finite element method- Help needed

    Dear all, I have written a code for dynamic analysis of a mechanical structure. My primary purpose is to find natural frequencies of the structure. When I test my code for a cantilever bar whose natural frequencies are known analytically, I found a big difference between the the first...
  40. H

    Stiffness matrix in finite element method

    Dear all, I have written a code for structural analysis using the finite element method. For some reason, I directly started with 3D elements ( hexahedron). I used to believe that the code was fine but recently i realized that the results ( deformation, natural frequency,..) strongly depend...
  41. H

    Natural Frequency in Finite Element Method

    Hi all, A fixed-free bar has a single natural frequency. When we discretize such a bar in the finite element method, then the natural frequencies are the eigenvalues and an nχn matrix where n is the number of the degree of freedom which is usually large. Thus we obtain up to n natural...
  42. H

    Dynamic Structural Analysis Using Finite Element Method

    Hi all, I need to do a dynamic structural analysis using finite element method and I have a question about the mass matrix. Question: I have the force per nodes and I need to calculate the displacement of each node at a given time. For this purpose, it seems that I need to distribute the...
  43. A

    Finite element method for shallow water equations

    Hello, I am trying to solve the shallow water equations using finite element method. Can anyone explain me how to treat nonlinear term in the Galerkin equation? so for example in the equation for the velocity we will have the term u\nabla v where u and v are the velocity components. For...
  44. B

    Finite Element Method vs. Integrated Finite Difference for Complex Geometries

    Hello all: For modeling flow (or whatever) in a non-rectangular geometry, can anyone comment on whether the finite element method would be better or worse or the same as the integrated finite difference method? I'm reading some papers by competing groups (so I can decide which code to...
  45. N

    Learning the Finite Element Method: A Practical Guide for Engineers

    Hi guys I wish to self-study the Finite Element Method. I am mostly interested in learning how to understand and implement the method rather than to investigate if a solution exists, i.e. I wish to follow the "engineering-approach" rather than the "mathematician-approach". Do you have any...
  46. S

    How Does MATLAB Implement Finite Element Method for Modified Poisson's Equation?

    Homework Statement The MATLAB program below uses the finite element method to solve a Dirichlet problem for Poisson's equation. -a1u''(x) = 0, xl < x < xh u(xl) = 0, u(xh) = 1 Modify the problem to solve -u''(x) + u(x) = 1, xl < x < xh...
  47. M

    Best Finite Element Method Books to Understand Methods

    do suggest a good book for finite element method?some books have left me confused over the various methods
  48. S

    Finite element method (proof question)

    Homework Statement this is part of a theorem for the error estimate for the model problem for finite element method. i have to prove the following inequality: \left|u'(x)-\tilde{u}'_h(x)\right|\leq max_{0\leq(y)\leq1}\left|u''(y)\right| Homework Equations \tilde{u_h} is the...
  49. L

    Solving Elasticity Problems with Finite Element Method

    I am trying to make a program that solves elasticity problems with finite element method and I don't understand how to bring in boundary conditions. Constant displacement boundary conditions seem simple: replace variables that represent the displacements at surface nodes with the prescribed...