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

    Perturbation Theory on Finite Domains

    In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode y'' + P(x)y = 0 using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question...
  2. J

    How to Expand Finite Series in a Shorter Form?

    Hello :blushing: How to do expand this: (\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2 where X(t_j)-X(t_{j-1}) = \Delta X_j to this: (\sum_{j=1}^{n}(\Delta X_j)^4 + 2*\sum_{i=1}^{n}\sum_{j<i}^{ }(\Delta X_i)^2(\Delta X_j)^2 -2*t*\sum_{j=1}^{n}(\Delta X_j)^2+t^2I get near the North Pole... but...
  3. B

    Rationalizing fractions over finite fields

    Homework Statement Let w be a primitive n-th root of unity in some finite field. Let 0 < k < n. My question is how to rationalize [\tex]\dfrac{1}{1 + w^k}[\tex]. That is, can we get rid of the denominator somehow? I know what to do in the case of complex numbers but here I'm at a loss...
  4. Jalo

    Finite square well potential energy

    Homework Statement Hello. Imagine a particle bound in a square well potential of potential energy V0 if |x| > a 0 if |x| < a The wave function of the particle is: (ignoring the time dependency) -A*exp(kx) if x<-a B*sin(3*pi*x/4a) if |x|<a A*exp(-kx) if x>a where k =...
  5. K

    Analytical solutions for electric field of finite rectangular sheet

    Hi, I have been trying to find analytical solutions for a finite rectangular sheet, say, in the xy plane, with dimensions a and b. Assume it is uniformly charged. An excellent (and short) description of the problem is here. The three integrals for Ex(x,y,z), Ey(x,y,z) and Ez(x,y,z) given on...
  6. S

    Finite Element Meshing: Understanding Node Intersections

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  7. Ryuzaki

    How are Natural Numbers Constructed from the Class of All Finite Sets?

    Class of all finite sets In a higher algebra book that I'm working through, the natural numbers are constructed in the following manner:- Consider the class S of all finite sets. Now, S is partitioned into equivalence classes based on the equivalence relation that two finite sets are...
  8. S

    What are the Symmetry Constraints for a Quarter-Shape Concrete and Steel Box?

    Hello, Please help my question is really basic,but i need help I have concrete box and steel box and bolt as illustrated in picture one(quarter shape),so my questions are: the key points in symmetry for the quarter of the shape are: 1.prevent translation in the two plan...
  9. Fernando Revilla

    MHB Total order relations on a finite set

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my complete response.
  10. D

    Interstellar travel is impossible, moore's law is finite, the heat dea

    Yeah, all of those things, multipled by five thousand, and there we have the current cosmic predicament for human beings. Is there any way in which we can do something about this? Or will Schopenhauer have the last laugh?
  11. M

    Potential of a finite line of charge

    Homework Statement A finite line of charge (L=L1+L2) with a linear density of d(x)=k.x, in which k>0. This finite line goes from -L2 to +L1 in the x axis. Calculate the electric field and the electric potential in the point P=(0,H). Homework Equations dV=(1/(4*pi*ε0))*dq/r The...
  12. L

    Question about finite dimensional real l^p space

    I believe I understand the definitition of the l^p space, its set of infinite sequences that converge when the sequence is put to the power of p, term by term. However I came across "Let T be a real linear operator from a finite dimensional real l^p space to a real finite dimensional banach...
  13. E

    Can the universe be flat, yet finite?

    I keep on reading that cosmologists contemplate two possibilities: either the universe is closed, unbounded and therefore finite, or else it is open (possibly flat) and infinite. I hear that a flat, finite universe "introduces many problems" and is discarded. My question would be "what...
  14. J

    Calculation mass of a finite universe

    Over the past few months, I researched how to validate the mass of ordinary matter in the universe assuming a finite volume. Three of my previous posts involved issues related to this question. Generally, the number E56 grams is quoted but without assumptions or calculations. I used one...
  15. S

    Finite size effects at first order phase transitions

    Hi. I am trying to understand some features related to the order of a phase transition. It is known that there are finite size effects in a finite system. The finite size scaling theory provides relations between some quantities with the length of the system L. At second order phase...
  16. T

    Group definition for finite groups

    Was wondering if the only required definition for finite groups is closure (maybe associativity as well). It seems that is all that is necessary. The inverse and identity necessarily seem to follow based on the fact that if I multiply any element by itself enough times, I have to repeat back to...
  17. C

    Finite Hilbert Space v.s Infinite Hilbert Space in Perturbation Theory

    Hi all, I have a question about the concept of complete set when I apply the perturbation theory in two situations -Finite Hilbert Space and Infinite Hilbert Space. Consider a Hamiltonian H=H0+H', where H0 is the unperturbed Hamiltonian and H' is the perturbed Hamiltonian. Let ψ_n be the...
  18. Petrus

    MHB Area finite region bounded by the curves

    Hello MHB, I got stuck on an old exam determine the area of the finite region bounded by the curves y^2=1-x and y=x+1 the integration becomes more easy if we change it to x so let's do it x=1-y^2 and x=y-1 to calculate the limits we equal them y-1=1-y^2 <=> x_1=-2 \ x_2=1 so we take the right...
  19. M

    MHB What Is a Finite Simple Group of Order Two?

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  20. S

    Finite Element MATLAB Code for Solving Boundary Value Problems

    Homework Statement Consider the problem $$-u''\left(x\right) = 1, \;\; 0 < x < 3, \;\; u \left(0\right) = 0, \; -u' \left(3\right) = u\left(3\right)+1.$$ Formulate a MATLAB code to produce the solution and plot the solution from 0 to 3. Homework Equations The Attempt at a Solution Multiply by...
  21. A

    Showing ADM angular momentum to be well-defined and finite

    Hello! I am trying to show that the ADM definition of angular momentum is well-defined and finite. Here is the definition: J^{i} = -\frac{1}{2}lim_{r\rightarrow\infty}\int_{S_{r}}\epsilon_{ijm} x^{j} (k_{mn} - \overline{g}_{mn} tr k) dS_{n} I'm working with an asymptotically flat...
  22. T

    Valid Basis Functions for Triangular 2D Elements: Solving for Coefficients

    Howdy, I am trying to formulate a proof to show that the shape function [N(x,y)] = [Ni(x,y) Nj(x,y) Nk(x,y)] and the the basis functions Ni(x,y) = (1/2A)(ai + bix + ciz) Nj(x,y) = (1/2A)(aj + bjx + cjy) Nk(x,y) = (1/2A)(ak + bkx + cky) are valid for triangular, 2 dimensional...
  23. Fernando Revilla

    MHB Marcus 's question at Yahoo Answers (Bijectivity on finite and infinite sets)

    Here is the question: Here is a link to the question: Abstract math question: bijectivity on finite and infinite sets? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  24. B

    Scattering from finite square barrier

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  25. A

    Quantum Mechanics - Finite Square Well - Graphical Solution

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  26. F

    Finite Differences Method for Physics

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  27. F

    Finite Element analysis: Free body diagram help

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  28. A

    Free Finite Element Analysis Software

    Anyone know of some good, free finite element analysis software? I used NASTRAN in college, but from what I can tell that costs at least several thousand.
  29. A

    Electric Field of Finite, Diagonal Line-Charge

    Homework Statement Consider the triangular-shaped wire in the picture. The base is of length 2a and is along the y-axis, and the two sides are of length 3a each. The wire is uniformly charged with charge density (per line) \lambda. Find the electrostatic field at all points \vec{x} = x\hat{i}...
  30. 7

    Energies and numbers of bound states in finite potential well

    Hello I understand how to approach finite potential well. However i am disturbed by equation which describes number of states ##N## for a finite potential well (##d## is a width of a well and ##W_p## is potential): $$ N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi} $$ I am sure it has something to...
  31. 7

    Plotting a wavefunction for a finite potential well doesn't work out

    Lets say we have a finite square potential well like below: This well has a ##\psi## which we can combine with ##\psi_I##, ##\psi_{II}## and ##\psi_{III}##. I have been playing around and got expressions for them, but they are not the same for ODD and EVEN solutions but let's do this only...
  32. J

    Comp. Physics: Finite time Carnot cycle

    Hi, For my Bachelor's thesis I've been working on a finite time Carnot cycle. I've finished my numerical analysis using the differential equations governing the time evolution. My next step should be a simulation. First I should stick to a 1 dimensional system. This system consists of a...
  33. T

    Composition Factors cyclic IFF finite group soluble

    Hey, just trying to get my head around the logic of this. I can see that if composition factors are cyclic then clearly the group is soluble, since there exists a subnormal series with abelian factors, but I am struggling to see how the converse holds. If a group is soluble, then it has a...
  34. A

    What does finite volume charge density mean?

    Homework Statement "as long as the volume charge density is finite (which is not true of surface charge distributions or point charges), the electric field is continuous. Homework Equations The Attempt at a Solution I know that for surface charges distributions and point charges...
  35. 7

    Finite Potential Well Solutions

    Lets say we have a finite square well symetric around ##y## axis (picture below). I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for the regions I, II and III. \begin{align} \text{I:}& & \psi_{\text{I}}&= Ae^{\kappa x} \\...
  36. caffeinemachine

    MHB Finite abelian group textbook help

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  37. E

    Suppose an electron was kept with an alpha particle at a finite

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  38. E

    A problem from finite element book

    ..though I figure it's sort of an analysis type problem. ∫wvdx=0 (int from 0 to 1) for all v in V. w is continuous on [0,1]. What it means to be in V: v in V satisfies being continuous on [0,1], v(0)=v(1)=0, and derivatives of v are piecewise continuous . Problem is: Show that w(x)=0 for x in...
  39. C

    Proof: Integral is finite (Fubini/Tonelli?)

    Homework Statement Let f:[0,1]→ℝ be non-negative and integrable. Prove that \int_{[0,1]}\frac{f(y)}{|x-y|^{1/2}}dy is finite for ae x in [0,1] Homework Equations This looks like a Fubini/Tonelli's Theorem problem from the problem givens. The Attempt at a Solution I honestly don't know...
  40. K

    Transmission phase of finite 1D photonic crystal

    I use transfer matrix method. I try to plot the phase Φ(ω) versus frequency ω graph in the vicinity of a band gap. The transmission coefficient t(ω) is t(ω) = |t(ω)|*exp(i*Φ(ω)) from which I get the transmission phase Φ(ω) = atan(t_imaginary(ω)/t_real(ω)). Since I assume that -pi/2 ≦ Φ ≦...
  41. L

    Is the universe large and finite, or infinite?

    Dear Sir, If we consider the Big Bang Hypotesis , the age of the universe and the rate of expansion ogf space, the Universe could be very large , but finite. But Prof Sean Carrol said , in a lecture , that there's a possibility of Infinite Universe. Please elaborate.
  42. T

    How can the universe be infinite, if it has a finite age?

    Morning everyone, I apologize for bringing up a topic that has probably been discussed to death here in the past. I've been reading the FAQ, and a few old threads about finite vs infinite universe, but I'm still struggling to grasp both of these ideas. I'd be really grateful if someone could...
  43. P

    MHB Integral closure in finite extension fields

    Let $K=\mathbb{Q}[\omega]$ where $\omega^2+\omega+1=0$ and let $R$ be the polynomial ring $K[x]$. Let $L$ be the field $K(x)[y]$ where $y$ satisfies $y^3=1+x^2$.Which is the integral closure of $R$ in $L$, why?
  44. Vahsek

    Finite sum formula for tangent (trigonometry)

    Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help...
  45. S

    My Proof of Structure Theorem for Finite Abelian Groups

    Hello! If anybody has a minute, I'd appreciate a quick look-through of my proof that a finite abelian group can be decomposed into a direct product of cyclic subgroups. I'm new to formal writing (as well as Latex) and all feedback is greatly appreciated! Thanks in advance for your time...
  46. A

    MHB Find the finite sum of the square and cube exponent of integers

    Hey, it is clear for me that \sum_{i=1}^{n} i = \frac{n(n+1)}{2} how to find a formula for \sum_{i=1}^{n} i^2 \sum_{i=1}^{n} i^3 Thanks
  47. C

    First Post: How to Smooth End point of Finite Data Series time series

    I wish it wasn't out of desperation that I'm making this first post! I have a neural network that is making predictions, the next 5 time points per training. Back testing consists of appending these 5 point sets together to produce a data set that spans time over a much longer period...
  48. R

    Let G be a finite group in which every element has a square root

    Homework Statement Let G be a finite group in which every element has a square root. That is, for each x in G, there exists a y in G such that y^2=x. Prove every element in G has a unique square root. Homework Equations G being a group means it is a set with operation * satisfying...
  49. S

    Maple How to plot and animate an ODE in Maple using finite difference scheme

    Hi, I am currently trying to plot and animate a wave function using the Schrodinger equation. I currently have the following finite difference equation:- i(\psi(x, t+\Delta t)-\psi(x,t))/(\Delta t)=-(1/2)*(\psi(x+\Delta x, t)+\psi(x-\delta x, t)-2*\psi(x,t))/((\Delta...
  50. Chris L T521

    MHB Unknownnn's question from Yahoo Answers (re: finite math/set theory)

    Here is the question: Here is a link to the question: Finite math problem involving venn diagrams? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
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