Finite Definition and 1000 Threads

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 Volume Method & Evaluating Integral at Borders for Two-Phase Flow

    Hi! I am trying to make a one-dimentional simulator for two-phase flow. I am going to use the finite volume method, because it is conservative and thus it's easier to keep track of the oil/water ratio in the area. Say you have a conservation equation on the form \nabla \cdot (k(x) \nabla P(x))...
  2. P

    What does the operator A'A represent in image processing?

    Hello all, I hope this is the write sub-forum for this question. I have been looking at the Laplacian of a 2-D vector field. It is explained nicely by this Wikipedia article here. My question is more regarding how these operators work together. So, in the case of the Laplacian, it tells me...
  3. Eagle9

    The number of laws of physics is finite or infinite?

    I think that it is relatively easy to simply count the number of physics that are aware for us as of February 2014. Probably there is statistics that deals with it and can tell us how many laws of physics exist now; maybe this number is equal to 1000, maybe more, I am not aware of it. But...
  4. S

    Least squares assumptions: finite and nonzero 4th moments

    This isn't a homework problem - I'm just confused by something in a textbook that I'm reading (not for a class, either). I'd appreciate an intuitive clarification, or a link to a good explanation (can't seem to find anything useful on Google or in my textbook). My book states that one of the...
  5. M

    MHB Implicit finite difference method

    Hey! :o I have a implicit finite difference method for the wave equation. At step 0, we set: $W_j^0=v(x_j), j=0,...,J$ At the step 1, we set: $W_j^1=v(x_j)+Dtu(x_j)+\frac{Dt^2}{2}(\frac{v(x_{j-1})-2v(x_j)+v(x_{j+1})}{h^2}+f(x_j,0)), j=0,...,J$ Can that be that at the step 1 $j$ begins from...
  6. 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...
  7. A

    Numerical boundary conditions for wide approximation finite difference

    Hi, I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is: u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2 or when discretized u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2 In addition to...
  8. V

    Work obtainable from two finite reservoirs

    The problem is as follows: A Carnot engine operates between two finite size reservoirs, one a body of water of mass MH at 100°C and the other a body of water of mass ML at 0°C. Find the maximum work obtainable from the two reservoirs. The Attempt at a Solution I haven't done...
  9. M

    MHB The finite difference method for the heat equation-error

    Hey! :o I am implementing in a program the finite difference method for the heat equation. The problem is the following: $$u_t(x,t)=(g(x,t)u_x(x,t))_x+f(x,t), \forall (x,t) \in [0,1]x[0,1]$$ $$u(0,t)=u(1,t)=0, \forall t \in [0,1]$$ $$u(x,0)=0, \forall x \in [0,1]$$ where $f(x,t)=\pi x...
  10. M

    How could nothing is finite yet unbounded?

    Hey guys. It's been awhile I don't posting in this thread. How could space which is nothing, is finite yet unbounded? Aren't beyond the so-called finite yet unbounded universe is just nothing that we called as space?
  11. E

    Solving Finite Series: \sum_{l=0}^{k-1} (r+l)^j (r+l-k)^i

    Hi! I've encountered the series below: \sum_{l=0}^{k-1} (r+l)^j (r+l-k)^i where r, k, i, j are positive integers and i \leq j . I am interested in expressing this series as a polynomial in k - or rather - finding the coefficients of that polynomial as i,j changes. I have reasons to...
  12. G

    Unbounded Hamiltonian leading to finite ground state

    If a Hamiltonian is unbounded from below, say the hydrogen atom where the Hamiltonian is -∞ at r=0, is there a way to tell if the ground state is bounded (e.g. hydrogen is -13.6 eV and not -∞ eV)? It seems if the potential is 1/r^2 or less, then the energy will be finite as: \int d^3 r (1/r^2)...
  13. GeorgeDishman

    Gravitational time dilation for a spherical body of finite radius

    I am considering the gravitational time dilation at the centre of a spherical, non-rotating body (such as the Earth). The usual formula for gravitational time dilation is √(1-r_s/r) where r_s is the Schwarzschild Radius and r is the radius of the clock compared to one at infinity, however, this...
  14. K

    Deterministic Finite Automata - Myhill Nerode

    Homework Statement Let ## f\left( b_{1}, \dots , b_{n} \right) ##be a boolean function. Define ##S_{f} = \{\left( b_{1}, \dots , b_{n} \right): f\left( b_{1}, \dots , b_{n} \right)=1; b_{i} \in \{0,1\}, 1\leq i \leq n \}##The subsets ##S_{f}## are viewed below as languages consisting of...
  15. C

    Electric field (at any point) due to a finite cylinder

    Homework Statement I'm not even attempting the graph yet, but I'm having trouble figuring out how to do this problem for a finite cylinder. All I've found in my notes is finite spheres and infinite cylinders. Homework Equations E=∫[ρdv]/4∏εR2] \hat{R} The Attempt at a Solution...
  16. J

    Trace of elements in a finite complex matrix group is bounded

    Homework Statement Let G be a finite complex matrix group: G \subset M_{n\times n}. Show that, for g \in G, |\text{tr}(g)| \le n and |\text{tr}(g)| = n only for g = e^{i\theta}I. 2. The attempt at a solution Since G is finite, then every element g \in G has a finite order: g^r = I for some...
  17. djh101

    Exploring the Finite and Infinite Solutions of ODEs with k < 1/4

    Consider \frac{d^{2}y}{dx^{2}}+\frac{k}{x^{2}}y = 0. Show that every nontrivial solution has an infinite number of positive zeroes if k > 1/4 and a finite number if k ≤ 1/4. Solving gives: y = Asin(\sqrt{k}ln(x)) + Bcos(\sqrt{k}ln(x)) And setting y = 0 gives: tan(\sqrt{k}ln(x)) =...
  18. T

    Infinite universe from a finite start

    Hello, I have a question about the character of the universe today and its early state. As I understand it there is no consensus as to whether the universe today (the whole universe, not the observable) is infinite, finite, or finite but looped in on itself. It seems to me to follow that if...
  19. 3

    Equivalence Relations, Cardinality and Finite Sets.

    Hey everyone, I have three problems that I'm working on that are review questions for my Math Final. Homework Statement First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z | x - y =5} and find the equivalence classes. Is Z | R a partition? Homework...
  20. 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...
  21. 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...
  22. DreamWeaver

    MHB Finite Tangent product / quotient

    Just for fun, eh...? (Heidy)For z \in \mathbb{R}, and m \in 2\mathbb{N}+1, show that:\frac{\tan mz}{\tan z}=\prod_{j=1}^{ \lfloor m/2 \rfloor } \tan\left(\frac{j\pi}{m}+z\right) \tan\left(\frac{j\pi}{m}-z\right)
  23. D

    How to quantify fringing fields around a finite width plate electrode?

    Homework Statement This isn't homework or coursework as such, but i thought it may be the best place to ask this question. The last time i posted in the other section it was deleted! Im considering the case of an electrode of finite width L in the x direction. The y direction is...
  24. M

    MHB Finite difference method-convergence

    Hello! :) I am implementing the finite difference method in a program in C and I got stuck at the rate of convergence.. The formula is \frac{log(\frac{e_{1}}{e_{2}})}{log(\frac{J_{2}}{J_{1}})} , right? where e_{i}=max|y_{j}^{J_{i}}-y(x_{j}^{J_{i}})| , 0<=j<=J_{i} . How can I find the J_{1}...
  25. E

    Using finite difference method for solving an elliptic PDE with MATLAB

    Homework Statement Given that we the following elliptic problem on a rectangular region: \nabla^2 T=0, \ (x,y)\in \Omega T(0,y)=300, \ T(4,y)=600, \ 0 \leq y \leq 2 \frac{\partial T}{\partial y}(x,0)=0, \frac{\partial T}{\partial y}(x,2) = 0, \ 0\leq x \leq 4 We want to solve this problem...
  26. O

    Bound state of finite square well, why do we make this statement?

    Reading from http://quantummechanics.ucsd.edu/ph130a/130_notes/node150.html Again we have assumed a beam of definite momentum incident from the left and no wave incident from the right. Why is the above statement made? What does the reflected wave mean? There is now all why reflected...
  27. T

    Isomorphic Finite Dimensional Vector Spaces

    I'm going through the text "Linear Algebra Done Right" 2nd edition by Axler. Made it to chapter 4 with one problem I'm unable to understand fully. The theory that two vector spaces are isomorphic if and only if they have the same dimension. I can see this easily in one direction, that is...
  28. F

    Proving Finite Squares in n!+n^p-n+2 for p=3,5 (mod8)

    Homework Statement How to prove the following: Let p be a prime p=3,5 (mod8). Show that the sequence n!+n^p-n+2 contains at most finitely many squares. Should I build a contardiction or prove it directly? I really need some help 2. The attempt at a solution Use Fermats...
  29. 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...
  30. H

    Universe finite or infinite during the first instance of inflation?

    -In the first few fractions of a second after the big bang, was the universe finite and closed during early inflation, before it smoothed out and became flat and infinite? I am wondering because I would like to know if the theory implies that the universe initially inflated with a finite...
  31. 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
  32. S

    Does the vacuum of space have finite electrical resistance?

    Air typically has a very high but non zero resistance. Given that air is just a medium, and that space is also just a medium, does the vacuum of space have a fundamental constant of electrical resistance, or is the electrical resistance of space truly infinite? How is this proven one way or the...
  33. S

    Five point scheme Finite Difference Method

    For possion equation $$u_{xx}+u_{yy}=f$$ I know the general five point scheme is in the form $$a_{1}U_{i,j-1}+a_{2}U_{i-1,j}+a_{3}U_{i,j}+a_{4}U_{i+1,j}+a_{5}U_{i,j+1}=f_{i,j}$$ But , is there have the form...
  34. O

    Lowest energy state with infinite and finite potential

    Hello everyone and thanks for reading my post. I have a problem with an electron, which actually is confined into a region 0 ≤ x≤ L with infinite potential around it, and its energy in the ground state is 0.38eV. Then on the x > L region the potential is 5eV and the energy of the lowest...
  35. R

    Determinant of a Finite Field 2x2 Matrix

    Homework Statement Find the determinant of: |1 1| |2 1| The field is Z3. Homework Equations The field is Z3, that is, to multiply two numbers, you first multiply then take the remainder of the division by 3. The Attempt at a Solution I tried: ( 1 x 1 ) - ( 1 x 2 ) 1 x 1 will...
  36. D

    MHB Infinite domain to finite plate by a change of variables

    Consider the following solution to the steady state heat diffusion problem on an infinite y domain. \[ T(x, y) = \sum_{n = 1}^{\infty}c_n\exp\left(-\frac{\pi n}{\ell} y\right) \sin\left(\frac{\pi...
  37. A

    Question regarding Pulse shape's effect on the finite slope of GM-tube

    Hi, I am hoping someone here could help me understand the finite slope of the counting plateau in Geiger Muller Tubes. Master Knoll says this, "In real cases, the counting plateau always shows some finite slope, as shown in Fig. 7.5b. Any effect that adds a low-amplitude tail to the...
  38. B

    Finite Fourier Transform on a 3d wave

    Finite Fourier Transform on a 2d wave How does the finite Fourier transform work exactly? The transform of f(x) is \widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for z(x,y,t)=X(x)Y(y)T(t)...
  39. H

    Convert differential equation to finite difference equation

    I have the differential equation \frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt} which is the first term from M(t)=4\pi\int_0^{r(t)}C(r,t)r(t)^2dr This describes the change in mass (M) of a sphere from a change in radius (r) given a density (rho) that depends on radius and time (t). My...
  40. T

    Shift in Mercury's Perihelion by finite light speed

    Someone published a simple computation of the relativistic shift in Mercury's perihelion (over and above classical, ie. the small correction over the classical-mechanical shift) by more or less using the principle of relativity. I believe it was a she and she computed how far mercury travels...
  41. Math Amateur

    MHB Finitely Generated k-algebra - Nature of the finite generation - basic question

    In Dummit and Foote Chapter 15 on page 657 we find the following definition of a k-algebra: Let k be a field. A ring R is a k-algebra if k is contained in the centre of R and the identity of k is the identity of R. This defintion is followed by the definition of a finitely generated k-algebra...
  42. N

    Kramers-Kronig relations on a finite data set

    Hi Say I have a finite data set (frequency, absorption) and I would like to find the corresponding dispersion. For this I could use the Kramers-Kronig (KK) relation on the absorption data. What I would do is to make a qubic spline and then perform the KK-transformation. However, the absorption...
  43. D

    How can you accurately calculate fault current on a finite bus by hand?

    so calculating FC on an infinite bus is easy but how do you calculate it by hand for a finite bus? here is the concept i am having an issue understanding: let say i have a transformer that is 225KVA with 5%z and 3 phase 480 to 208. with infinite bus i have 12491A available on the...
  44. M

    Finite dimensionality of a Discrete time system.

    Homework Statement Determine the finite dimensionality of the following system: y[n] = nx[n] Homework Equations y[n]= f(y[n−1], y[n−2],..., y[n−N],x[n],x[n−1],..., x[n−M],n) Where N is how many dimensions the system has. The Attempt at a Solution I understand that the following system...
  45. 7

    FINITE potential step - SOS - got lost in system of equation

    Homework Statement Homework Equations I know that energy mentioned in the statement is kinetic energy so keep in mind when reading that ##E\equiv E_k##. In our case the kinetic energy is larger than the potential energy ##\boxed{E>E_p}## and this is why stationary states for the regions 1...
  46. G

    The axiom of choice one a finite family of sets.

    The axiom of choice on a finite family of sets. I just been doing some casual reading on the Axiom of CHoice and my understanding of the is that it assert the existence of a choice function when one is not constructable. So if we have a finite family of nonempty sets is it fair to say we can...
  47. E

    Electric field of finite line charge and 2 point charges

    Homework Statement A finite uniform linear charge ρ_L = 4 nC/m lies on the xy plane; start point and end point are (7,0,0) and (0,7,0) .While point charges of 8 nC each are located at (0, 1, 1) and (0, -1, 1). Find E at (0, 0 ,0) Homework Equations dE=ρ_L *dz'/4∏ε *...
  48. 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...
  49. S

    Is a finite semigroup isomorphic to subsets of some group?

    Is any given finite semigroup isomorphic to some finite semigroup S that consists of some subsets of some finite group G under the operation of set multiplication defined in the usual way? (i.e. the product of two subsets A,B of G is the set consisting of all (and only) those elements of G that...
  50. 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.
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