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. ElijahRockers

    Transforming Finite Series: Solving with Z-Transform?

    Homework Statement Let ## x_j = \begin{Bmatrix} {1, 0 \leq j \leq N-1} \\ {0, else} \\ \end{Bmatrix} ## Show that ##\hat{x}(\phi) = \frac{e^{-i\frac{N-1}{2}\phi}sin(\frac{N}{2}\phi)}{sin(\frac{1}{2}\phi)}##Homework Equations [/B] ##\hat{x}(\phi) := \sum_{j = -\infty}^{\infty} x_j...
  2. F

    A question on the commutativity of finite rotations

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  3. F

    Electric field on a point around a finite wire

    Homework Statement In the figure below, there is a linear rectilinear uniform wire with charge density of ## \lambda ##. It its located at the Z axis, where z1 = a and z2 = b, (b> a) The point O is the origin of the coordinates. "R" is the cylindrical polar radial coordinate. a) Find the...
  4. Z

    How can I solve two related non-linear equations using EDP finite volume method?

    Hi evryone, the system i am working on, is composed of two relatied non-linear equations that I discretised using finite volume method with fully implicit scheme, the difficulty is that the two unknonw appear in both equations and i don't know how to solve it. Any propositions? Thank you.
  5. Last-cloud

    Finite difference method nonlinear PDE

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  6. L

    Unit solid angle and finite angle

    Homework Statement A point source emits visible light isotropically. Its luminous flux is 0.11 lumen. Find the flux whithin the cone that has half angle of 30 degree from the light source. Homework Equations luminous flux = luminous intensity * solid anlge The Attempt at a Solution I tried...
  7. gracy

    Why a finite angular displacement is not a vector?

    One of my friend has answered this question in this way. Angular displacement can't be a vector because addition is not commutative. Say we are looking at the Earth with North America facing us and the North Pole facing up:if we rotate the Earth so that we move 90 degrees north, now the NP is...
  8. S

    Finite solutions of Brocard’s problem

    x^2=n!+1⇒ (x+1)(x-1)=n! where (x+1)/2 and (x-1)/2 are consecutive integers and have consecutive primes as factor ,let ,y and z (respectively) so it can be written y-1=z. Consider prime counting function π(z),π(2z-1) that count primes less than the variable or argument. It can be seen that f(z)...
  9. A

    Acceleration of electron due to finite sheet at voltage

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  10. M

    MHB Number of states of finite automata

    Hey! :o Prove that for each $n>0$, a language $B_n$ exists where $B_n$ is recognizable by an NFA that has $n$ states, and If $B_n=A_1 \cup \dots \cup A_k$, for regular languages $A_i$, then at least one of the $A_i$ requires a DFA with exponentially many states. Could you give me some...
  11. P

    Finite T transverse magnetization of transverse Ising chain

    Homework Statement Consider the transverse field Ising model, with $$H=-J\sum_i\left(\sigma^x_i\sigma^x_{i+1}+g\sigma^z_i\right)$$ I have to calculate the magnetization $$\langle\sigma_z\rangle$$ at finite temperature. Homework EquationsThe Attempt at a Solution I have to say, I'm a bit lost.
  12. gfd43tg

    Finite square well ##\psi(x)## solution for ##-a < x < a##

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  13. H

    Measurability of a function with finite codomain

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  14. B

    Finite square well potential numerical solution

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  15. T

    Help understanding Non-determinate Finite Automaton

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  16. A

    Why do people think the universe is finite?

    Why do they asume that the big bang is the origin of the universe? While the big bang might have occurred, it is not the "origin" of the universe. It at most is the origin of the expansion of matter through the universe. Think about the universe as a huge infinite vacuum that contains matter...
  17. C

    Finding the Potential of a Charged Rod on the x-axis

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  18. D

    Formula for Helmholtz Coil with a finite thickness?

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  19. Z

    What is the finite expansion of a function means ?

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  20. M

    MATLAB Matlab finite difference schemes

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  21. evinda

    MHB Show $\bigcup A$ is Finite When $A$ is a Finite Set of Finite Sets

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  22. evinda

    MHB Could the Cartesian Product be finite?

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  23. evinda

    MHB Each subset of the natural numbers is finite or countable

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  24. evinda

    MHB The subsets of finite sets are finite sets.

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  25. A

    Automorphism Group of Radical of Finite Group

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  26. Greg Bernhardt

    Challenge 25: Finite Abelian Groups

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  27. A

    Subnormal p-Sylow Subgroup of Finite Group

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  28. W

    Finite and infinite unitary transformation

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  29. D

    Einstein's argument for a finite universe

    After reading some of Einstein's writings on relativity I am confused as to why he finds it necessary that the universe is finite. In his "Relativity", Einstein explains that the ultimate Newtonian picture of the universe (matter in Euclidean space) would be one in which all mass were...
  30. Fallen Angel

    MHB Finite Group Inverses: Proving $N_{ABC}=N_{CBA}$

    Hi, I bring a new algebraic challenge ;) Let $G$ be a finite group and $U,V,W\subset G$ arbitrary subsets of $G$. We will denote $N_{UVW}$ the number of triples $(x,y,z)\in U\times V \times W$ such that $xyz$ is the unity of $G$, say $e$. Now suppose we have three pairwise disjoint sets...
  31. O

    Finite difference discretization for systems of higher ODEs

    How can I use finite difference to discretize a system of fourth order differential equations? for example: y(4)+5y(3)-2y''+3y'-y=0
  32. feynwomann

    Solve Finite Potential Well: Schrödinger Eqn. & k=qtan(q*a)

    ' I've got these solutions to the Schrödinger equation (##-\frac{\hbar} {2m} \frac {d^2} {dx^2} \psi(x) + V(x)*\psi(x)=E*\psi(x)##): x < -a: ##\psi(x)=C_1*e^(k*x)## -a < x < a: ##\psi(x)=A*cos(q*x)+B*sin(q*x)## x > a: ##\psi(x)=C_2*e^(-k*x)## ##q^2=\frac {2m(E+V_0)} {\hbar^2}## and ##k^2=\frac...
  33. M

    Center of a group with finite index

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  34. W

    Determining A Finite Value of Infinity

    okay...if you accept that the sequence 1+2+3+4...=-1/12, I think I have determined a finite value of infinity. To find the value of the sums of all natural numbers up to a number, you can use the equation ((x^2)+x)/2. An example would be 4. 4+3+2+1=10. ((4^2)+4)/2 also equals 10. following...
  35. Math Amateur

    Free Modules: Bland Corollary 2.2.4 - Issue on Finite Generation

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4. Corollary 2.2.4 and its proof read as follows: In the second last paragraph of Bland's proof above we read: " ... ... If...
  36. Math Amateur

    MHB Free Modules: Solving Issue of Finite Generation Corollary 2.2.4

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4. Corollary 2.2.4 and its proof read as follows: In the second last paragraph of Bland's proof above we read: " ... ... If...
  37. caffeinemachine

    MHB Proof Checking: Submodules of a Free Module of Finite Rank over a PID is Also Free of Finite Rank

    I have been trying to prove the following: Let $R$ be a P.I.D. and $M$ be a free module of finite rank over $R$. Then every submodule $N$ of $M$ is a free module with $$\text{rank} N\leq \text{rank} M$$ and here is my "proof": Using the fact that $R$ is a P.I.D., the theorem is trivial...
  38. J

    MHB Convert a non-deterministic finite automata to a regular expression.

    Hi, I'm trying to covert a NFA to a regular expression and I've manged to come up with an answer but I don't think that it is right. Here's the question - http://i.imgur.com/NUHxTXY.png And here's my workings -...
  39. H

    Bose Einstein condensation in 2D finite space

    It can be easily proved that Bose Einstein condensation can be got in infinite 2D. But what about finite 2D with extreme large "Volume" L^2 ?
  40. M

    Finite Difference Expressed As a Probability Generating Function

    $$F(z) = \sum_{n=0}^\infty a_n x^n $$ $$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$ So, we can begin to piece together some differential equations in terms of generating functions in order to satisfy some discrete recursion relation (which is the desired problem to solve). However I...
  41. I

    How Does a Particle Behave in a 2-Level Finite Potential Well?

    Consider a particle of mass m subject to the following potential function (taking Vo and L to be positive): V (x) = 40 Vo if x < 0; 0 if 0 < x < L/2; 2 Vo if L/2 < x < L; 40 Vo if x > L. (a) Derive the transcendental equation for energy eigenstates having an energy 2 Vo < E < 40 Vo. To simplify...
  42. P

    What Is the Probability of Particle Ionization in a Shifted Finite Square Well?

    Homework Statement Consider a particle of mass m in the ground state of a potential well of width 2 a and depth. the particle was in the ground state of the potential well with V0 < Vcritical, which means the well is a narrow one. At t = 0 the bottom of the potential well is shifted down to Vo'...
  43. P

    Canonical Commutation Relations in finite dimensional Hilbert Space?

    So lately I've been thinking about whether or not it'd be possible to have the commutation relation [x,p]=i \hbar in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the...
  44. Math Amateur

    MHB Modules of Finite Length - Cohn, page 61

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of the length of a module. Some analysis follows, as does a statement of Theorem 2.5. I need help to understand...
  45. B

    Finite Temperature Density Matrix Calculation

    Homework Statement Consider the Hamiltonian ##H=\begin{bmatrix} 0& \frac{-iw}{2}\\ \frac{iw}{2} & 0 \end{bmatrix}## Write the finite temperature density of the matrix ##\rho(T)## Homework Equations ##\beta=\frac{1}{kT}##The Attempt at a Solution The initial part of the problem had me find the...
  46. M

    Electric field along a finite rod

    Homework Statement Homework Equations V = q/(4*pi*E_0*r), when 0 is taken at infinity dV = -E*dsThe Attempt at a Solution a. The total charge of the rod is given by Q = lambda*L So the potential at P is given by V = (lambda*L)/(4*pi*E_0*y) b. We can calculate the electric field by...
  47. A

    The finite size of the nucleon

    Hi, I read in article: to incorporate the effects of the finite size of the nucleon, we considered an exponential form factor. I want to know what does "the finite size" mean? thank you
  48. Math Amateur

    MHB Regular representations of finite dimensional algebras

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we read the following on page 57: https://www.physicsforums.com/attachments/3149I am trying to gain an understanding of representations. I would...
  49. T

    Physics Project using finite method?

    I'm supposed to be doing a project. Here is what it says to do. I tried to copy and paste the directions in here but some of the equations are not turning out on this page as expected, so I have uploaded the project. Can anyone tell me how I need to start or what to do please? The...
  50. Chacabucogod

    Solving the Finite Element Method Matrix with Rao - Engineering

    I was reading the finite element method in engineering by Rao and in the first example he ends up with a matrix that is singular. The matrix is the following: \begin{pmatrix} 2 &-2 & 0\\ -2 & 3&-1\\ 0&-1& 1 \end{pmatrix} Which is a symmetric matrix as far as I can remember...
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