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. Euge

    POTW Is the Finite Expectation of Powers Satisfied by Nonnegative Random Variables?

    Suppose ##X## is a nonnegative random variable and ##p\in (0,\infty)##. Show that ##\mathbb{E}[X^p] < \infty## if and only if ##\sum_{n = 1}^\infty n^{p-1}P(X \ge n) < \infty##.
  2. jjson775

    I Why do exponential functions occur outside of finite walls?

    Between the walls of a finite well, the solution to the time independent Schrodinger equation is a combination of sines and cosines. Outside the walls where E - Uo is positive, the solutions are exponential functions. Why?
  3. person123

    Python My Python Library For Finite Difference Method

    I recently made a Python library for modelling (very basic) finite difference problems. The Github readme goes into details of what it does and how it works, and I put together a Google Colab with some examples (diffusion, advection, water wave refraction) with interactive visuals. I'd love to...
  4. Euge

    POTW Product of Two Finite Cyclic Groups

    For each positive integer ##m##, let ##C_m## denote a cyclic group of order ##m##. Show that for all positive integers ##m## and ##n##, there is an isomorphism ##C_m \times C_n \simeq C_d \times C_l## where ##d = \operatorname{gcd}(m,n)## and ##l = \operatorname{lcm}[m,n]##.
  5. yashabyadav

    A Need help implementing finite difference method for BPM

    I am trying to apply finite difference scheme for Beam propagation method by following this paper. I was wondering if anyone can share their code if they have implemented this method. I can share my code which is not working as expected and can get some insights if possible.
  6. P

    I Finite many Lattice Points in Sphere?

    Hello, I am wondering if in an n-ball the number of lattice points is finite. First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...
  7. Coelum

    I Finite Well Problem: Can Energy Be Equal to V0?

    Dear PFers, I have a question about the finite potential well problem. Let's assume the well is centered in 0 and the potential is V0 outside the well and 0 inside. For more details see Wikipedia: https://en.wikipedia.org/wiki/Finite_potential_well. Now, the question is: can the energy of the...
  8. gratusri

    Optimizing and minimization of a Deterministic Finite Automata

    so this is the question , I have to minimize this DFA this is How I did it but when I checked for answers , this is what it was, can someone please explain to me what mistake I made? I have been wondering about this for past 2 days
  9. Ashish Somwanshi

    Finite potential well problem penetration depth

    I don't understand where I went wrong, the formula and calculations which I have attached are correct...please do help if anyone can spot the mistake.
  10. 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...
  11. Ashish Somwanshi

    Maximizing Learning: The Importance of Showing Your Work in Problem Solving

  12. 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...
  13. S

    I Operators in finite dimension Hilbert space

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  14. 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...
  15. M

    I Polynomials can be used to generate a finite string of primes....

    F(n)=##n^2 −n+41## generates primes for all n<41. Questions: (1) Are there polynomials that have longer lists? (2) Is such a list of polynomials finite (yes, no, unknown)? (3) Same questions for quadratic polynomials?
  16. Graham87

    Quantum mechanics - finite square well

    In a) I get that T should be largest where V_0 is least wide, because when V_0 is infinitely wide the particle would be fully reflected. But I don't get how height in b) and energy levels height in c) correlates to T and R. Is it because of their k? I get the opposite answer from the correct...
  17. D

    Understanding a question about a finite energy well when E < 0

    So ,solution only fit for one of them , the other one is not zero , how can that be solution?? I am pretty new to quantum physics..
  18. 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...
  19. F

    What's the Optimal Shape Function for Phase Field Simulations in FEM?

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

    A Finite difference scheme for y'(t)=a*y(t)

    It seems no finite difference scheme is stable for a>0, dt>0, correct?
  21. angweieng

    A Definition of finite size, radiative and weak magnetism corrections

    Hi all, I would like to understand the definition of finite size correction, radiative correction and weak magnetism correction, with their impacts on the beta spectrum. I'm not a physics student, thus I would like to seek for a help about the simple explanation that can be understand by...
  22. Buzz Bloom

    I Calculating the probability that the Universe is finite

    Reference https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf I note that the use of Gaussian probabilities is mentioned many times in the reference. However in many discussions via posts in many threads, there seems to be a consensus that the distribution is...
  23. Buzz Bloom

    I Please check my calculation of the radius of a finite universe

    My source for Ωk and H0 is https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf , page 40, equation 47b. Ωk = 0.0007. Page 15 equation 13 has H0 = 67.27 km/(s Mpc). The formula I used is R = (c/H0) (1/Ωk)1/2 . ( I did have a reference for this, but I misplaced it, and I...
  24. Buzz Bloom

    I A relationship between flat vs. finite and (2) Hubble constant tension

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  25. alan123hk

    B Why is Light Speed Invariant & Finite?

    I think infinite speed is unimaginable. If something is moving at infinite speed, we can't find it at all because it has moved to infinity. Furthermore, when the maximum speed is limited, a reasonable inference should be that observers in different reference frames should find the same one speed...
  26. L

    Potential due to a finite charged wire

    Considering a reference frame with ##x=0## at the leftmost point I have for the leftmost piece of wire: ##\int_{x=0}^{x=2R}\frac{\lambda dx}{4\pi\varepsilon_0 (3R-x)}=\frac{\lambda ln(3)}{4\pi\varepsilon_0}##. The potential at O due to the semicircular piece of wire at the center is...
  27. M

    Dynamic Systems: Poincaré-Bendixson Theorem finite # of equilibria

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  28. H

    A quite verbal proof that if V is finite dimensional then S is also....

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  29. KyleGranger

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    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!
  30. F

    Weak solutions under finite elements

    Finite elements give weak solutions, that is, the solutions to a PDE are only correct in its integral form. Is it possible that in finite element software, the solution differs a lot from the analytic one locally while it's exact in its integral form (globally)?
  31. F

    I Finite difference scheme for u'=u^2, u0=1, 0<t<2

    u'(t)=u(t)^2, u(0)=1, 0<t<2 What finite difference scheme can overcome the difficulty when t->1+ and help the solution jump to the negative branch?
  32. R

    Show that the sum of the finite limits of these two series is also finite

    In the homework I am asked to proof this, the hint says that I can use the triangle inequality. I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,
  33. F

    I Conservation of finite difference for vibration equations

    Let's discuss whether the energy under a finite difference (FD) scheme is conserved. Take the simplest vibration eq mx''+kx=0, which one will use a FD scheme to solve. The energy is mx'^2/2+kx^2/2. Whether the energy is conserved doesn't depend on the FD scheme for the ODE but upon the FD scheme...
  34. Paige_Turner

    B How can we watch BHs collide in a finite time?

    A BH can't touch another hole's horizon for the same reason nothing else can: time drags the object to a halt for a distant observer. Right? Manifestly not. So are we wrong about gravitational time dilation, or what?
  35. DanielJ2021

    I Infinite space in a finite bubble, in a Tegmark Level 2 multiverse?

    So I am a layman in physics, I admit I am trying to grasp big ideas piecemeal via articles, wikipedia and YouTube. I don't pretend to be educated in this regard but I am curious and willing to learn! The idea of the multiverse intrigues me. Sidestepping for a second the fact that the idea has...
  36. T

    A Noether's theorem for finite Hamiltonian systems

    The Noether's theorem for finite Hamiltonian systems says that: My question is: If I know a symmetry how can I write the first integral?
  37. 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?
  38. F

    Electrostatic energy in finite space

    Electrostatic energy involves a volume integral and a surface integral The question is how to apply this formula to a finite space in which case the 1st term (surface integral) won't vanish. Let's apply to a capacitor and enclose the capacitor by a closed surface. Calculate the energy integral...
  39. T

    A Photon self-energy in the finite temperature field theory (real -time)

  40. 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...
  41. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  42. B

    Finite bending of an elastic block - Equilibrium equations

    I am studying the finite bending of a rubber-like block, assuming Neo-Hookean response. In the following, ##l_0##,##h##, ##\bar{\theta}## are parameters, while the variables are ##r## and ##\theta##. The Cauchy stress tensor is ##T= - \pi I + \mu(\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r \otimes...
  43. A

    I Determine if polynomial system has finite number of solutions

    Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states: I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have...
  44. BWV

    B What would you pay to play (or avoid) a finite St Petersburg game?

    The St Petersburg Paradox is a game with an infinite expected value, but the 'paradox' is that a rational person would pay a relatively small sum (maybe $20) to play it (see https://en.wikipedia.org/wiki/St._Petersburg_paradox ) So say the government offered you the game with a tax-free payout...
  45. cianfa72

    I The spacetime length of finite spacelike intervals

    Hello, I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR. All us know the physical meaning of timelike paths: basically they are paths followed through...
  46. S

    I The adjective "finite" applied to algebraic structures

    The adjective "finite" applied to many algebraic structures (e.g. groups, fields) indicates a set with a finite number of elements. However, (as I understand it) "finite algebra" refers to a finitely generated algebra. Are there other examples where "finite" means finite in some respect but...
  47. D

    Calculate the Energy Levels of an Electron in a Finite Potential Well

    Thank you for reading :bow: Section 1 To find the energy states of the particle, we define the wave function over three discrete domains defined by the sets ##\left\{x<-L\right\}##, ##\left\{-L<x<L\right\}##, and ##\left\{L<x\right\}##. The time independent Schrodinder equation is...
  48. rcgldr

    Factoring polynomials with finite field coeffcients

    I'm not sure if I should post this here or in the mathematics section. I'm trying to find a way to implement a mapping of a larger finite field such as GF(2^64) to a composite field GF((2^32)^2). Let f(x) be a primitive polynomial for GF(2^64), with 1 bit coefficients. If the coefficients of...
  49. D

    Understanding the Energy States of a Particle in a Finite Potential Well

    To find the energy states of the particle, we define the wave function over three discrete domains defined by the sets ##\left\{x<-L\right\}##, ##\left\{L<x\right\}## and ##\left\{|x|<L\right\}##. The time independent Schr\"odinder equation is...
  50. T

    What could be some future developments of Explicit Finite Method?

    Hello everyone, I have to submit an assignment by Monday and there is only one part of the assignment missing, which is future developments of the Explicit Finite Difference Method. However, I can not do it because I really do not know what exactly he is asking me for on there. Could someone...
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