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. T

    What are the limitations of the Explicit Finite Difference Model?

    Homework Statement:: Discuss the limitation of the Explicit Finite Difference Model. Relevant Equations:: no formula Hello there, I have to discuss the limitations of using the Explicit Finite Difference model to calculate a 2D Heat Diffusion through an aluminium place, however, I really...
  2. T

    Finite Square Well, Ψ[SUB]III[/SUB] const related too Ψ[SUB]II[/SUB]?

    I'm following Griffith's Modern Physics 2nd edition chapter 5. I got to the part where we make ΨI(0) = ΨII(0) I get that αCeα(0) = QAsin(Q(0)) - QBsin(Q(0)) => C = QA/α But when I try to graph it, the region I distribution doesn't seem to equal the region II distribution at 0. The book goes...
  3. kmm

    I Observing Black Holes in Finite Time

    My understanding from General Relativity is that if as distant observers we watch a probe or any test mass approach a black hole, time dilation goes to infinity as the probe gets closer to the event horizon. This would imply that we would never observe a black hole form, or the collision of two...
  4. I

    I Symmetry and Finite Coupled Oscillators

    For an infinite system of coupled oscillators of identical mass and spring constant k. The matrix equation of motion is \ddot{X}=M^{-1}KX The eigenvectors of the solutions are those of the translation operator (since the translation operator and M^{-1}K commute). My question is, for the...
  5. D

    A Numerical Hartree Fock with Finite Difference Matrices for Helium

    Here is the paper again: https://www.mdpi.com/2218-2004/6/2/22?type=check_update&version=2#related_content For a class project I need to calculate the energy levels of atoms using the Hartree Fock method as presented in this paper which essentially brute forces the calculation using finite...
  6. F

    Vector space of functions from finite set to real numbers

    Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...
  7. J

    I Decomposition per the Fundamental Theorem of Finite Abelian Groups

    According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders. Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group. Unfortunately, the book gives no...
  8. A

    I Finite Lorentz Transformation via Poisson Bracket

    Let me define ##L_{x;v}## as the operator that produce a Lorentz boost in the ##x##-direction with a speed of ##v##. This operator acts on the components of the 4-position as follows $$L_{x;v}(x) =\gamma_{v}(x-vt),$$ $$L_{x;v}(y) =y,$$ $$L_{x;v}(z) =z,$$ $$L_{x;v}(t)...
  9. K

    I Finite fields, irreducible polynomial and minimal polynomial theorem

    I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...
  10. jaketodd

    I Equation for Gravity Going to Zero at Finite Distance?

    Seeking an equation for gravity where gravitational force goes to zero at large distances I realize the Newtonian formula has trouble with this. And I've heard gravity never goes to zero in relativity. So, maybe a quantum gravity one, that isn't too complicated? Thanks!
  11. L

    I Symbol for Finite: Exploring the Use and Interpretation of ##<\infty##

    I know there's a symbol for infinite. But is there a finite one?
  12. P

    MHB Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

    In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this...
  13. Terrycho

    I The wave function in the finite square well

    Hello! I have been recently studying Quantum mechanics alone and I've just got this question. If the potential function V(x) is an even function, then the time-independent wave function can always be taken to be either even or odd. However, I found one case that this theorem is not applied...
  14. dk123

    Proof that a projectile comes to rest in a finite distance

    equation of motion : while
  15. E

    Gauss' law for a physical capacitor with finite thickness plates

    In derivations of capacitance it is standard to consider two oppositely charged, infinitely thin sheets. If we construct a Gaussian cylinder across one sheet, we obtain ##E_{1} = \frac{\sigma}{2\epsilon_{0}}## for one sheet, and then we can superpose this field with that from the other at an...
  16. evinda

    MHB Interval with Dirac function in a finite interval

    Hello! (Wave) I want to calculate the integral $\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$. I have done the following so far: $$\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^1\sin \left (\pi (t-1)\right )\delta (-t+1)\...
  17. W

    Engineering Derivation of the error due to a finite load attached to a potentiometer

    x = fraction of potentiometer connected to load Vp in parrallel with VL = x/(Rp/RL.x.(1-x) + 1) If RL = infinite, then Ro = x and Vo = x.Vs If RL = finite, then Ro = x/(Rp/RL.x.(1-x) + 1) and Vo = x.Vs/(Rp/RL.x.(1-x) + 1) Therefore error is x.Vs - x.Vs/(Rp/RL.x.(1-x) + 1) Trying to break the...
  18. jeremiahrose99

    I Solution to the 1D wave equation for a finite length plane wave tube

    Hi there! This is my first post here - glad to be involved with what seems like a great community! I'm trying to understand the acoustics of a finite plane-wave tube terminated by arbitrary impedances at both ends. So far all of the treatments I've managed seem only to address a different...
  19. D

    Finite rotations and infinitesimal rotations

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  20. TopologyisGeometry

    Plotting the radial wave function of Deuteron in a finite well

    To plot ##u(r)## we need to find the solutions for each region. Which is in the relevant equations part. Now, I have to do this numerically. Using python 3.7 I made an ##u## which is filled with zeros and a for loop with if/elseif statement, basically telling it to plot values for whenever...
  21. 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...
  22. I

    Problem on a set which is a subset of a finite set

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  23. C

    I Finite expansion of a fraction of functions

    I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...
  24. L

    Finite difference method for the diffusion-advection equation

    So for my scheme I obtained ##\frac{\mu}{h^2} U_{p}+(\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{E}+(\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{N} - (\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{W} - (\frac{v_{2}}{2 h} + \frac{\mu}{h^2})U_{N} + \tau = f## however I am not sure this is correct. I am quite new to...
  25. 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##...
  26. K

    I Finite density at the BB preinflation epoch implies a finite universe?

    http://astronomy.swin.edu.au/cosmos/B/Big+Bang D (density) = m/V At t = 10^-47, D = 10^93 kg/cm ^3, r = 10^-57 meters, V = 4pi(r^3)/3 which is about (4/3)pi(10^-171)(meters^3) t = 10^-47 precedes the inflationary epoch at t = 10^-35, this is important since this implies all matter at the big...
  27. C

    B How can a dimension be "curled up" and have a finite extent?

    As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I...
  28. L

    A Representations of finite groups: Irreducible and reducible

    Matrix representation of a finite group G is irreducible representation if \sum^n_{i=1}|\chi_i|^2=|G|. Representation is reducible if \sum^n_{i=1}|\chi_i|^2>|G|. What if \sum^n_{i=1}|\chi_i|^2<|G|. Are then multiplication of matrices form a group? If yes what we can say from...
  29. L

    A Representations of finite groups -- Equivalent representations

    I am confused. Look for instance cyclic ##C_2## group representation where D(e)= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} and D(g)= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} and let's take invertible matrix A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}. Then A^{-1}=...
  30. 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...
  31. Saptarshi Sarkar

    I Effects of photons having a finite mass

    Few days back, our college professor told us that if a photon were to have a finite mass, then the Coulomb potential between two stationary charges separated by a distance r would be strictly zero beyond some distance. He told us that it was due to the reason that photon is the elementary...
  32. S

    B Exploring the Finite Universe Concept: Insights from the Big Bang Theory

    My first question. I was taught in school about Big Bang as a theory of physics, so then if the entire content of our universe came out from one tiny point boundary over a finite span of time, would that mean the content was finite?
  33. T

    Finite quantum well, multiple choice question

    Homework Statement:: Consider an electron trapped in a one-dimensional finite well of width L. What is the minimum possible kinetic energy of the electron? A) 0 B) Between 0 and h^2/8mL^2 C) ≈h^2/8mL^2, but it is not possible to find the exact value because of the uncertainty principle D)...
  34. Ehyeh Asher Ehyeh

    I Is Our Universe Finite or Infinite?

    Summary: No answer could be more important to the assumptions and approach to cosmology. The overwhelming bias is a finite Universe, and could this be a mistake? The measurements across the observable universe strongly indicate a Gaussian Curvature of Zero(Flat). Does this prove that Spacetime...
  35. Haynes Kwon

    I Probability flux inside the finite potential barrier

    In Bransden textbook, it is stated that the probability current density is constant since we are dealing with 1-d stationary states. It gives probability flux outside the finite potential barrier which I verified to be constant with respect to x, but it doesn't provide the probability current...
  36. R

    Comp Sci Finite state machine that doesn't terminate and detects overlapping strings

    Attached is what I have so far. I believe it is done but I am not 100% sure. It seems to me like every case is considered. For each state, and output of a,b, or c is possible.
  37. Telemachus

    Magnetostatics and Ampere's law on a finite length wire

    I tried to think why Ampere's law seems to fail in this case. For me it was clear that there is no symmetry in the z direction, there is no translational symmetry because of the finiteness of the wire. On the other hand, I know that Ampere's law is independent of the loop we take. This also...
  38. J

    Biot-Savart for finite volume currents (Wangsness)

    Since ##\vec{R}/R^3 = -\nabla(\frac{1}{R}) = \nabla^\prime(\frac{1}{R})##, the standard form of the Biot-Savart law for volume currents can be re-written as: $$\frac{\mu_0}{4\pi}\int\limits_{V^\prime}\frac{\vec{J}^\prime (\vec{r}^\prime)\times\vec{R}}{R^3}d\tau^\prime =...
  39. C

    MHB Finite vs Ring Groups: Examining Theorems

    Dear Everyone, Does every theorem that holds for finite group holds for ring groups? Why or Why not?Thanks Cbarker1
  40. Yalanhar

    Calculate the electric field due to a line of charge of finite length

    Homework Statement: A thin rod of length L and charge Q is uniformly charged, so it has a linear charge density ##\lambda =q/l## Find the electric field at point where is an arbitrarily positioned point. Homework Equations: ##dE=\frac{Kdq}{r^2}## A thin rod of length L and charge Q is...
  41. I

    Simple Finite Square Well Problem help *Ignore, made stupid mistake*

    I've tried to carry out the solution to this as a normal 2nd order Differential Equation ##\psi ##'' - ##-k^2 \psi ## = 0 Assume solution has form ##e^{\gamma x}## sub this in form ##\psi## and get ##\gamma ^2## ##e^{\gamma x} ## + ##k^2 e^{\gamma x}## = 0 Solution is ##\gamma## = 0 or ##k^2##...
  42. Philip Koeck

    A What is the mean inner potential in an ideal crystal of finite size?

    In a hypothetical, electrically neutral, ideal crystal, where all unit cells are identical, even the ones at the surface: What would the average value of the electrostatic potential be compared to that of the vacuum outside the crystal? Would it be the same or more positive? As a simple example...
  43. M

    I What if the Jacobian doesn't exist at finite points in domain of integral?

    Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...
  44. 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...
  45. patrykh18

    I Numerical solution to the Schrodinger eqn. using Finite Difference Method

    As part of my project I was asked to use the finite difference method to solve Schrodinger equation. I see how you can turn it into a matrix equation, but I don't know how to solve it if the energy eigenvalues are unknown. Are there any recommended methods I can use to determine those...
  46. Buzz Bloom

    What is the Mean Expected Volume of a Finite Universe?

    Problem Statement Assumptions: a. The universe is finite. That is, it is (approximately) a 3D boundary of a 4D hyper-sphere of radius r. b. [The following is based on https://arxiv.org/pdf/1502.01589.pdf as discussed in the thread...
  47. E

    B Finite & Infinite: Univerally Accepted Definitions?

    I'm troubled by what I think the 'community' considers them to be, but I'm not sure if I'm correct. It appears as though finite is thought to have both an end and a beginning, but is it true that infinite (infinity) is thought to only have no end? Is this accurate? If so, then it would seem like...
  48. nomadreid

    Computational learnability: jump from finite to infinite

    Summary: Starting on basics of computational learnability, I am missing the key intuition that allows results about finite processes to reference results from the infinite domain. Snowball: (a) Hazard brought my attention to a popular article...
  49. 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...
  50. mertcan

    I AICc value derivation (Akaike information criteria for finite samples)

    Hi everyone, initially let me introduce a concept widely used in ARIMA in the following. $$AICc = AIC + \frac {2k^2+2k} {n-k-1}$$ where n denotes the sample size and k denotes the number of parameters. Thus, AICc is essentially AIC with an extra penalty term for the number of parameters. Note...
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