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. O

    Finite Difference Methods and Global Error

    I was going through my notes on different finite difference methods and came across something I don't quite understand. I have code that will calculate an approximate solution we can call this U_nm that I define on a grid using h and dt for the change in x and time respectively. Now I have...
  2. P

    Symmetery of a finite sequence of numbers

    Hi All; I attach a pdf file on something I have been working on for some time. Any feedback would be appreciated. Regards Garbagebin
  3. S

    Finite square well potential question

    For a finite one-dimensional square potential well if a proton is bound, how many bound energy states are there? If m = 1.67*10^(-27) kg a = 2.0fm and the depth of the well is 40MeV. Now I know the energy levels are En = (n^2 * h^2) /(8ma^2) = (n^2*pi*2)/4 * (2hbar^2)/(ma^2) but I am...
  4. C

    Proving Finite Index Subgroups in G Have Normal Subgroups of Lower Index

    Homework Statement Prove: If H is a subgroup with finite index in G Then there is a normal subgroup K of G such that K is a subgroup of H and K has index less than n! in G. Homework Equations Note: |G:H| represents the index of H in G |G:H| is the number of left cosets of H in G, ie...
  5. F

    If f is meromorphic on U with only a finite number of poles, then

    If f is meromorphic on U with only a finite number of poles, then f=\frac{g}{h} where g and h are analytic on U. We say f is meromorphic, then f is defined on U except at discrete set of points S which are poles. If z_0 is such a point, then there exist m in integers such that (z-z_0)^mf(z)...
  6. J

    Any way to figure out what this finite geometric series sums to?

    I would like to find a nice formula for \sum_{k=0}^{n - 1}ar^{4k}. I know that \sum_{k=0}^{n - 1}ar^{k} = a\frac{1 - r^n}{1 - r} and was wondering if there was some sort of analogue.
  7. M

    Explicitly describing the singular locus from a finite set of polynomials

    When explicitly given a set of polynomial equations, I am interested in describing its singular locus. I read this from several sources that a point is singular if the rank of a Jacobian at a singular point must be any number less than its maximal possible number. Or is it the locus where all...
  8. Z

    Irreducible polynomial over finite field

    Homework Statement Factor x^16-x over the fields F4 and F8 Homework Equations factored over Z (or Q), x^16-x = (x*(x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) The Attempt at a Solution I know the that quadratic and higher terms I have left...
  9. J

    Set of real numbers in a finite number of words

    Hello everybody, Yesterday I've read that there exist a real number r which cannot be defined by a finite number of words. This result, although quite awesome, is so strange that it lead Poincaré to doubt Cantor's work and state "never consider objects that can't be defined in finite number...
  10. T

    Very Simple Question About Finite Differences/Sequences & Series

    Alright, so I just started my Calculus class, and we're doing Series and Sequences. We've been doing all these problems, and as I was doing them, I noticed that they all can be solved using the finite differences method. Why bother with sequences and series formulas when I can just use the...
  11. N

    Electric field due to a FINITE cylinder of charge - Tricky binomial expansion

    1. Homework Statement (a) Calculate the electric field at an axial point z of a thin, uniformly charged cylinder of charge density ρ , radius R, and length 2L. z is the distance measured from the center of the cylinder. (b) What becomes of your result in the event z >> L ? 2. Homework...
  12. D

    A question regarding finite potential wells

    Hi guys! This is my first post on Physics Forums even, and I have a question regarding potential wells with finite potential. I understand the infinite potential well but what if the well is finite? For example, if we a potential well with infinite potential to the left of 0, but with increasing...
  13. D

    Electric field of a uniform finite cylinder

    Homework Statement I have a solid cylinder of uniform charge density whose axis is centered along the z-axis. I am trying to calculate the electric field at a point on the z-axis. What I'm trying to do is to start by first calculating the field of a disk centered on the z-axis at a point on...
  14. A

    Depth of a finite square potential problem

    Homework Statement Consider a finite square-well potential well of width 3.00x10-15 m that contains a particle of mass 1.88 GeV/c2. How deep does the well need to be to contain three energy levels? Homework Equations The Attempt at a Solution I think I have to use the formula for penetration...
  15. B

    The limit of finite approximations of area

    My textbook never mentioned what happens when you multiply something by infinity. I would think 4 * ∞ would be ∞. So to me that whole equation should simplify to 1 - ∞ which is ∞. I don't see how they get 2/3
  16. J

    Amperes circuital law for finite length of wire

    Why is that amperes circuital law gives the same magnetic field around a finite legnth of wrie as if it is an infintie legnth of wire? By biot-savarts law we know that for a finite length of wire magnetic field is μ i ( cos θ1 - cos θ2)/ 2∏r I searched this question in google and one of...
  17. B

    Who does Finite Element Modelling?

    Hi Could anyone point me in the right direction? I'm employed at a small firm, only three employees at the time being. Our competencies spans from architecture to programming. We're in need of somebody who can help us create a finite element solution which we can implement in the cad...
  18. M

    Explaining Finite Solvable Groups: Understanding Burnside's Theorem

    HI, I was reading an article and it says that a finite group of order p^aq^b, where p, q are primes, is solvable and therefore not simple. But I can't quite understand why this is so. I do recall a theorem called Burnside's theorem which says that a group of such order is solvable. But then I...
  19. I

    Set of finite subsets of Z+ is denumerable

    Hi I am trying to prove that P=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\;X\mbox{ is finite }\} is denumerable. Now here is the strategy I am using. Let A_n=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\; |X|=n\;\} So A_n are basically sets of subsets of \mathbb{Z^+} with cardinality n. So...
  20. M

    Derivative of Finite Sums: Solve Your Problem Here

    I have problem like attached.
  21. G

    Finite Intersection of Open Sets Are Always Open?

    Suppose we have non-empty A_{1} and non-empty A_{2} which are both open. By "open" I mean all points of A_{1} and A_{2} are internal points. There is an argument -- which I have seen online and in textbooks -- that A_{1} \cap A_{2} = A is open (assuming A is non-empty) since: 1. For some x...
  22. O

    Fourier transform limit of finite signal

    So this is a very simple question that I am having some trouble figuring out: Let s(t) be a finite energy signal with Fourier Transform S(w). Show that \lim_{w \to \infty } S(w) = 0 We know by defintion that the FT of this signal is \ints(t)e^{-jwt}dt and also that ∫|s(t)|2dt < ∞. I'm a...
  23. H

    Nonlinear PDE finite difference method

    Hello I want to resolve a nonlinear partial differential equation of second order with finite difference method in matlab. the equation is in the pdf file attached. Thanks
  24. D

    Cardinality of the set of all finite subsets of [0,1]

    Hello, I was wondering this, what is the cardinality of the set of all finite subsets of the real interval [0,1] It somehow confuses me because the interval is nonnumerable (cardinality of the continuos \mathfrak{c}), while the subsets are less than numerable (finite). It is clear that it has...
  25. S

    When to use Order Notation? (Error in Finite differences)

    Homework Statement I'm having a hard time understanding when we approximate higher order powers by order notation, especially when it comes to working out the Truncation Error for Finite Differences. My notes say "We use the order notation O(h^{n}) and write X(h) = O(h^{n}) if there exists a...
  26. 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...
  27. C

    Is a finite function with finite Fourier transform possible?

    Clarification: I have seen in quantum mechanics many examples of wavefunctions and their Fourier transforms. I understand that a square pulse has a Fourier transform which is nonzero on an infinite interval. I am curious to know whether there exists any function which is nonzero on only a...
  28. I

    Difference between Central Difference Method and Finite Difference Method

    Hello all, I am in the process of solving a finite elements problem involving obtaining deflection of a simple mass-spring-damper 2nd order ODE system with a defined forcing function. While going through my class notes, I came across the idea of the central difference method, which is...
  29. B

    SUBSET K of elements in a group with finite distinct distinct conjugates

    WTS, is that such set is a subgroup. I need to show closure under group operation and inverse. I can do the inverse which is usually the hardest part, but I'm stuck on the grp op. So let a in K and b in K, both have finite distinct conjugates. Their conjugates are in the group too. WTS...
  30. C

    Finite Difference Numerical Solution to NL coupled PDEs

    I have a system of non-linear coupled PDEs, taken from a paper from the 1980s which I would like to numerically solve. I would prefer not to use a numerical Package like MatLab or Mathematica, though I will if I need to. I would like to know if anyone knows how to solve non-linear coupled...
  31. fluidistic

    Rectangular finite potential well problem

    Homework Statement An electron enters in a finite rectangular potential well of length 4 angstroms. When the entering electrons have a kinetic energy of 0.7 eV they can travel through the region without having any reflection. Use this information to calculate the depth of the potential well...
  32. P

    Finite Element and CFL condition for the heat equation

    I am solving the heat equation in a non comercial C++ finite elements code with explicit euler stepping, and adaptive meshes (coarse in the boundaries and finer in the center). I am aware the CFL condition for the heat equation depends on dt/h**2 for the 1D, 2D, 3D case. When I solve the...
  33. F

    Deterministic Finite State Automaton Construction

    Homework Statement Find a simple DFA (i.e. deterministic finite automaton) that accepts all natural numbers n for which n mod 3 = 0. Hint: A natural number is divisible by 3 if its checksum (or sum of digits) is divisible by 3. Homework Equations The Attempt at a Solution...
  34. S

    Help Needed: Simplifying Finite Product - Can You Help?

    Hello Guys I could not simplify the finite product bellow to another expression I hope I get an answer from you guys Thank you
  35. E

    Finite difference and Runge-Kutta for PDEs

    I made a small program to simulate the time development of a 1D wavepacket obeying the Schrodinger equation, mostly in order to learn a new programming language - so in order to not have to invoke big numerical methods packages, I opted for the simplest solution: The standard three-point...
  36. E

    Inherent negativity of seemingly symmetric finite integer sets

    Hi everyone. My first post on this great forum, keep up all the good ideas. Apologies if this is in the wrong section and for any lack of appropriate jargon in my post. I am not a mathematician. I have a theory / lemma which I would like your feedback on:- Take a finite set S of integers which...
  37. T

    Reaching the Rindler horizon in a finite proper time

    Hi, I am trying to show that timelike geodesics reach the Rindler horizon (X=0) in a finite proper time. The spacetime line element is ds^{2} = -\frac{g^{2}}{c^{2}}X^{2}dT^{2}+dX^{2}+dY^{2}+dZ^{2} Ive found something helpful here...
  38. S

    Finite difference approximation for third order partials?

    I'm attempting to perform interpolation in 3 dimensions and have a question that hopefully someone can answer. The derivative approximation is simple in a single direction: df/dx(i,j,k)= [f(i+1,j,k) - f(i-1,j,k)] / 2 And I know that in the second order: d2f/dxdy(i,j,k)= [f(i+1,j+1,k)...
  39. B

    How Do Functions Converge in L^t Spaces on Finite Measure Domains?

    Homework Statement I have a sequence of functions converging pointwise a.e. on a finite measure space, \int_X |f_n|^p \leq M (1 < p \leq \infty for all n. I need to conclude that f \in L^p and f_n \rightarrow f in L^t for all 1 \leq t < p. Homework Equations The Attempt at a Solution...
  40. E

    Every finite domain is a division ring

    I am taking a first course in algebra and I am having issues with a detail in this proof that every finite domain is a division ring. The argument that I used is that (because of cancellation in domains) left & right multiplication by a nonzero element r in a domain R gives a bijection from R...
  41. 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...
  42. E

    Sum of a finite exponential series

    Homework Statement Given is \sum_{n=-N}^{N}e^{-j \omega n} = e^{-j\omega N} \frac{1-e^{-j \omega (2N+1)}}{1 - e^{-j\omega}}. I do not see how you can rewrite it like that. Homework Equations Sum of a finite geometric series: \sum_{n=0}^{N}r^n=\frac{1-r^{N+1}}{1-r} The Attempt at a...
  43. V

    Finding backward finite difference approximation to derivatives

    Problem - Find backward finite difference approximations to first, second and third order derivatives to error of order h^3 Attempt By Tailor’s series expansion f(x-h) = f(x) - h f’(x) + h^2/2! f’’(x) - h^3/3! f’’’(x) + … Therefore, f’(x) with error of order h^3 is given by f(x-h) = f(x)...
  44. J

    Entropy and Maximum work for two idential, finite sized bodies

    Two idential, finite sized bodies of constant volume and constant heat capacity are used to drive a heat engine- heat is taken from the hot (Th) body, work is done, and heat is ejected to the cold (Tc) body. Both bodies wind up at Tf (a) What is the change in the entropy of the system? (b)...
  45. K

    Apostol: infinity as finite point

    I found a torrent online of Apostol's "Mathematical Analysis" 1st edition and I think I found a typo, or whoever scanned the book cut off the edge a bit... Apostol writes that the extended real number system R* is denoted by [-∞, +∞] while the regular real number system R is denoted by (-∞...
  46. N

    Prove that this finite set is a group

    Homework Statement Let G be a nonempty finite set with an associative binary operation such that: for all a,b,c in G ab = ac => b = c ba = ca => b = c (left and right cancellation) Prove that G is a group. 2. The attempt at a solution Let a \in G, the set <a> = {a^k : k \in N} is a finite...
  47. V

    How Do You Apply Orthonormality and Completeness in Quantum Finite Square Wells?

    Homework Statement 1. Mixed Spectrum The finite square well has a mixed spectrum or a mixed set of basis functions. The set of eigenfunctions that corresponds to the bound states are discrete (call this set {ψ_i(x)}) and the set that corresponds to the scattering states are continuous...
  48. P

    McLaurin Expansion of finite sum

    Would you please find the McLaurin expansion of the following series to help me: M Ʃ Binomial(m + q - 1,q) [(a x)^q /((a x + b)^(m + q)] q=0 where M , m ℂ N^+; a, b > 0; MANY THANKS FOR YOUR HELP.
  49. K

    Is a number preceding infinity, finite?

    Hi, I'm not sure if this is the right section, but I'm talking about numbers :). The questions is as written in the title: Is a number preceding infinity, finite?
  50. E

    Defining a Signal. periodic, bounded finite etc.

    Im having a little trouble about how to go about defining this signal. It has a sqrt(-1) in it raised to a power so this is where i get confused. No doubt my poor algebra skills may be holding me back from understanding this problem. The signal is x(k)=j^-k u(k) I need to determine: A...
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