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. B

    Exponentials or trig functions for finite square well?

    How do you know when to use exponentials and trig functions when solving for the wave function in a finite square well? I know you can do both, but is there some way to tell before hand which method will make the problem easier? Does it have something to do with parity?
  2. evinda

    MHB How is the Finite Difference Method Applied in Wave Approximation?

    Hello! (Wave) We consider the finite difference method for the approximation $\left\{\begin{matrix} -u''(x)+q(x)u(x)=f(x)\\ u'(a)=u'(b)=0 \end{matrix}\right.$ and let $K$ be the $(N+2) \times (N+2)$ matrix of the method. Let $v \in \mathbb{R}^{N+2}, v=\begin{pmatrix} v_0\\ v_1\\ \dots\\...
  3. L

    Understanding the Finite Value of ##\zeta(-1)##

    \sum^{\infty}_{n=1}\frac{1}{n^{\alpha}}=\zeta(\alpha) For ##\alpha=-1## ##\zeta(-1)=-\frac{1}{12}## I do not see any difference between sum ##1+2+3+4+5+...## and ##\zeta(-1)##. How the second one is finite and how we get negative result when all numbers which we add are positive. Thanks for...
  4. H

    Finite differencing on non-uniform grids

    Hi, Recently I had to find a derivative on a uniform grid. Being naive I tried the following scheme: f'(x_{n})=Af(x_{n+2})+Bf(x_{n+1})+Cf(x_{n})+Df(x_{n-1})+Ef(x_{n-2}) Then write the f(x_{n\pm i}) in terms of f^{(n)}(x_{n}) by use of Taylor's theorem. This lead to a system of linear...
  5. I

    Finite difference method to solve first-order, multivariable

    I'm trying to replicate the model presented in this [paper](http://www.sciencedirect.com/science/article/pii/S1359431103000474), which is basically to model heat and mass transfer along a one-dimensional duct. There are four characteristic equations for this problem : Momentum conservation...
  6. evinda

    MHB How to Solve a Boundary Value Problem Using the Finite Difference Method?

    Hello! (Wave) I want to solve numerically the following boundary value problem: $\left\{\begin{matrix} -u''+qu=f & , x \in [a,b]\\ -u'(a)+d_1 u(a)=0 & \\ u'(b)+d_2 u(b)=0 & \end{matrix}\right.$ where $q(x) \geq 0 \forall x \in [a,b], d_1, d_2 \geq 0$. We consider the uniform partition of...
  7. naima

    Can Klein-Gordon Equation Solutions Have Compact Support?

    I am interested in the solutions of the Klein Gordon equation. Plane waves solutions are well known in physics. they look like ## e^ { i (kx - \sqrt{k^2 + m^2} t)}## or superpositions of them. They are finite when t or x go to infinity. I am looking for the general solution of the problem. In...
  8. T

    Heat transfer through finite temperature difference

    Hello, Heat transfer through finite temperature difference is known a irreversible process because heat cannot be transferred from cold to hot temperature without doing any additional work. But, how this transfer affect the efficiency of the system? How this heat transfer decrease the amount of...
  9. mariem.makh

    Fortran FORTRAN Code for Control Volume Finite Element Method Fluid Flow Problems

    where i can found an example A FORTRAN control volume finite element method code FOR FLUID FLOW PROBLEMS thanks for help
  10. A

    Finite difference Schrodinger equation

    I am simulating electrons inside a cylindrical well like the one shown on the first figure. My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation. To do so...
  11. M

    Why Does Increasing Final Run Time Cause Divergence in Finite Differencing?

    hi PF! I've attached pictures to help you all see what is happening. Basically, I am running a forward time-centered space finite difference scheme, which is $${h_i^{j+1} \approx \left[ h_i^j \left( \frac{h_{i+1}^j-2h_i^j+h_{i-1}^j}{\Delta z^2} \right) + \frac{1}{2}\left(...
  12. N

    Finite Difference Method for non-square grid

    Hi, I have written some codes for the finite difference solution of diffusion equation (\frac{\partial c}{\partial t}= D {\nabla^2 c}, where c is the species concentration and D is the diffusion coefficient) as follows: DO k= 1, tsteps+1 DO i = 2, zsteps DO j = 2, rsteps...
  13. A

    Finite difference Poisson's equation

    I am trying to solve the following eigenvalue differential equation numerically: ∇2ψ = Eψ , where the coordinate system is polar coordinates and the boundary condition is ψ(R,Φ)=0, where R is the radius of the disk i am working on. To solve it I am using a finite difference scheme, but there...
  14. D

    Finite Field Structure: Prime Order Cyclic Group

    Take a prime order cyclic group. I want to take that as the additive group of a finite field. Since every finite field of the same order is isomorphic to one another, does the isomorphism define a multiplicative group structure on my cyclic group elements?
  15. D

    Finite field with hard discrete log for both groups

    If there a finite field where both group structures have hard discrete logs? Discrete log in the additive group means multiplicative inverse.
  16. L

    MHB What is the Rank of the Direct Sum of Torsion-free Groups?

    If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...
  17. M

    Finite Difference Approach for a Moving Boundary Problem

    Hi PF! I was wondering if anyone could help me with a finite difference question? The problem I am doing is a 1-D space and time problem, so ##z## (space variable, from left to right) and ##t## (time) are my independent variables and my dependent variable is ##h##, the height, governed by a PDE...
  18. C

    Can You Help With Finite Element Analysis in Cylindrical Coordinates?

    I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
  19. S

    Finite Element Symmetry Problem

    Hi, I am running a finite element on a cylinder with that converges at the bottom for a opening, which is symmetrical in both directions so i modeled one quarter but the problem is my stresses are the same with when i compare with a full model that i also done but the deflections are different...
  20. P

    Finite tidal forces at black hole event horizon redux

    What's the best way to explain why tidal forces for an observer free-falling through an event horizon are finite? My first thought was to say that "gravity isn't a force, it's a curved space-time". On further thought, however, it seems to me that consideration of the Rindler horizon shows...
  21. G

    Electric field at point between two finite charged wires

    Homework Statement Two parallel charged wires are in vacuum. Width of wires is equal to the distance between them. Calculate electric field in the middle (point A). Homework Equations Superposition The Attempt at a Solution Using superposition, y components on vector E are cancelled. I get...
  22. newjerseyrunner

    Would an infinite universe has a finite diameter?

    I know that space is expanding, so the further away you go from my location, the faster space is expanding, asymptomatically approaching the speed of light. I also know that as relative velocities approach the speed of light, the length of space contracts. From this I come up with a limit for...
  23. M

    Finite Differencing Dynamic Boundary

    Hi PF! I'm using a finite differencing scheme to solve the following $$h_t = h h_{zz} + 2h_z^2$$ where the subscripts denote partial derivatives. The difficulty I'm facing is the boundary conditions are dynamic, and move with time ##t##. This makes choosing a ##\Delta z## very difficult and...
  24. fricke

    Particle in a box with the finite depth

    For particle in a box with the finite depth, is it traveling wave? or standing wave? I am confused with its ability to pass through the potential walls that is classically forbidden area which makes me think it is traveling wave. But for particle in a box with infinite potential, I understand...
  25. F

    Finite Element Analysis: PE of an element

    Homework Statement The potential energy of an element in terms of local displacements q1 and q2 is given by the expression: Π = 3(q1^2) - 6q1q2 + 9(q2^2) + 9q1 Write down the expressions for element stiffness matrix k and element force f. I am at a complete loss as to what...
  26. H

    Prove that a finite set with cancellation laws is a group

    If G is a finite set closed under an associative operation such that ax = ay forces x = y and ua = wa forces u = w, for every a, x, y, u, w ##\in## G, prove that G is a group. What I attempted: If we can prove that for every x ##\in## G, x##^{-1}## is also ##\in## G, then by the closure of the...
  27. A

    Fluid problem - periodic forcing over a finite region

    Hello, I am working on a solo project outside my domain of expertise (Physics PhD student). I am trying to analyze/replicate the wave phenomena shown in the following video: To summarize what I am doing: I need to analyze a simple (cylindrical) pool, say 17.5" wide, 4" deep Figure out how...
  28. C

    Virtual work in finite plane bending of Euler-Bernoulli beam

    Please refer to the following image, which shows a portion of the deformed centerline of a beam in its equilibrium configuration with a uniformly distributed load. The stress resultants are the axial forces T, transverse shears Q, and bending moments M at sections 1 and 2, with the rotations...
  29. M

    Finite Differences: Central vs Forward Scheme

    Hi PF! I am looking at finite differencing schemes and it seems we need more initial information to compute central finite differencing than forward finite differencing. Is this true, or am I understanding the process wrong? Thanks!
  30. C

    How many topologies exist on 4 points? Any nomenclature?

    Just for fun, I tried enumerating the topologies on n points, for small n. I found that if the space X consists of 1 point, there is only one topology, and for n = 2, there are four topologies, although two are "isomorphic" in some sense. For n = 3, I I found 26 topologies, of 7 types. For n...
  31. P

    Proof concerning the union of a finite collection of events

    Homework Statement Prove that[/B] P(\cup_{i=1}^n E_i) \geq \max_i P(E_i) (1) for n≥1 Homework Equations I know that P(\cup_{i=1}^n E_i) \leq \sum_{i=1}^n P(E_i). The Attempt at a Solution I know when n=1, trivially P(E_1) \geq \max_1 P(E_1) =P(E_1). So I was hoping I could use induction to...
  32. B

    Injection from finite set to equally sized set is surjection

    This is a rather simple question, so it has been rattling my brain recently. Consider a surjective map ## f : S \rightarrow T ## where both ## S ## and ## T ## are finite sets of equal cardinality. Then is ## f ## necessarily injective? I proved the converse, which turned out to be quite...
  33. ELB27

    Getting identity out of a finite number of permutaions

    Homework Statement Let ##P## be a permutation matrix. Show that for some ##N>0## P^N := \underbrace{PP...P}_{N \ \text{times}} = I 2. Relevant definitions A permutation matrix is a ##n\times n## matrix containing only zeros and ones such that there is exactly one ##1## per row and per column...
  34. Cluemore

    Transmission: Finite Potential Barriers & Potential Steps

    This may appear like a homework question, but I am not asking for answers for the question, so please don't remove this post! This is a conceptual question, and I just want to show how I came to that question. The following question, " An electron and a proton of identical energy E encounter...
  35. W

    "Minimal Cover" in Finite Collection of Sets?

    Hi All, Say we have a finite collection ## S_1,...,S_n ## of sets , which are not all pairwise disjoint , and we want to find the minimal collection of the ## S_j ## whose union is ## \cup S_j ## . Is there any theorem, result to this effect? I would imagine that making the ## S_j##...
  36. Feldman Sia

    Backward Finite Difference Heat Equation error

    I had these code in this forum but comes out error as below, any suggestion? Error 1 error C4430: missing type specifier - int assumed. Note: C++ does not support default-int c:\users\username\documents\visual studio 2010\projects\fdm 001\fdm 001\explicit 001.cpp 27 Error 2...
  37. N

    Integral Form of Gauss' Law at Center of Finite Wire

    At the exact center of a finite wire (i.e. a distance, say $L/2$ from each end), why can I not apply Gauss's Law in integral form to find an EXACT solution for the electric field? At the center of the wire, $E$ is entirely radial, so it seems like I should be able to draw an infinitesimally...
  38. gfd43tg

    Bound states in finite spherical well

    Homework Statement Homework EquationsThe Attempt at a Solution for ##r \le a## and ##l = 0##, the radial equation is $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - V_{0} = Eu $$ $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - [V_{0} + E]u = 0$$ call ##k^{2} = \frac...
  39. S

    Ampere's Law for a finite wire

    Greetings, I am working as a TA and I encountered a particular question which asks the student to use the Ampere's Law in order to get the magnetic field created by a semi-infinite wire. I know that there will be charge accumulation a time-dependent electric-field, hence a displacement current...
  40. N

    MATLAB 3D Finite different method using matlab

    Can anyone show me how to solve the 3D diffusion equation which has been modeled into FDM by using matlab?
  41. K

    MHB Basis Theorem for Finite Abelian Groups

    I am attempting to answer the attached question. I have completed parts 1-4 and am struggling with part 5. 5. Prove that if a^{l_0}b_1^{l_1}...b_n^{l_n}=e then a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e If |a|>|b1|>|b2|>...>|bn| then I could raise both sides of a^{l_0}b_1^{l_1}...b_n^{l_n}=e to the...
  42. K

    Is the mass of the universe finite (collection of objects)?

    Whenever I attempt to research this question, my search results yield "Is the Universe Infinite" where the question ALWAYS refers to the volume of the universe. This question is usually answered along the lines of: "If the universe is closed, than it's volume, aka it's 3D surface area in...
  43. S

    MATLAB Can You Solve This Finite Difference Equation Using Matlab?

    hi dear i have a question. i have equation (1/α ) dT/dt =d2T/dr2 +1/r dT/dr +d2T/dz T=T(r,z) T(Ri,z)=Ti T(Ro,z)=To T(r,0)=To dT(r,L)/dz =0 by finite difference method O(h^3) and this question's MATLAB program. is there anyone who can do it ? it is very important for me tnx
  44. H

    Gauss Law for finite line/plate

    Homework Statement I just noticed that whenever I'm doing a problem involving Gauss Law, it always involves an infinite line/plate. I can't seem to figure out why it must be infinite large/long. Here is an explanation I read, but don't quite understand...
  45. MidgetDwarf

    Applied Finite Math worth self learning or will I see the topics in future math courses?

    I google searched finite math after reading the course description in my community college course catalog. I am a math major and is fine math work learning or should I use my time wisely and learn other branches of math. Ie ode, pde, proof writing etc. Will my future classes cover some of the...
  46. R

    Can Any Finite Graph Have Vertices with Unique Edge Counts?

    Homework Statement Show that any finite graph contains two vertices lying on the same number of edges. Homework Equations None The Attempt at a Solution I am confused how my book proved this. Let G be a graph with n vertices ##v_1, ..., v_n.## Place ##v_i## in a pigeonhole labelled...
  47. homer

    SOLVED: Equipotential surfaces for finite line of charge

    Homework Statement Purcell 2.10 [/B][not the problem I'm asking about, but needed for Purcell 2.11 which I am asking about] A thin rod extends along the z axis from z = -d to z = d. The rod carries a charge uniformly distributed along its length with linear charge density \lambda. By...
  48. S

    An argument for "Brocard's problem has finite solution"

    Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard's_problem. According to Brocard's problem ##x^{2}-1=n!=5!*(5+1)(5+2)...(5+s)## here,##(5+1)(5+2)...(5+s)=\mathcal{O}(5^{r}),5!=k##. So, ##x^{2}-1=k...
  49. S

    Let [ ] be a countable number of finite sets. Prove [ ]

    Homework Statement Problem: Let A_1 , A_2 , . . . be a countable number of finite sets. Prove that the union S = ⋃_i A_i is countable. Solution: Included in the TheProblemAndSolution.jpg file. Homework Equations Set-theoretic algebra. The Attempt at a Solution Unless I missed something, it...
  50. Z

    Why the photoelectric absorption section finite at threshold

    I mean the photoelectric effect of the hydrogen atom. It is weird. By the Fermi golden rule, the transition or absorption rate is proportional to the density of the final states. At threshold, the electron has zero momentum and thus zero density of state. Therefore, the absorption coefficient...
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