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. Math Amateur

    MHB Finite Integral Domains .... Adkins & Weintraub, Proposition 1.5 ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as...
  2. Math Amateur

    I Finite Integral Domains .... Adkins & Weintraub, Propn 1.5

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as follows: At the end of the above proof...
  3. D

    MHB Proving Finite subgroups of the multiplicative group of a field are cyclic

    I am looking at this proof and I am stuck on the logic that $a^{p}$ = 1. For example, consider the group under multiplication without zero, ${Z}_{5}$, wouldn't 2^4 = 1 imply that the order is 4 not 5? We know that if G is a finite abelian group, G is isomorphic to a direct product...
  4. P

    I How is the universe expanding if the speed of light is finite?

    The universe seems to be expanding since the farther away an object is, the faster it is moving. However, because of the finite speed of light, the farther away we look in distance, the further back in time we look. Does that mean that galaxies were moving faster in the past and are now slowing...
  5. T

    Finite difference method; finding the 'Stencil'

    Homework Statement Determine the Finite Difference Method stencil for approximating a second derivative u''(x) at a discrete set of nodes with maximum accuracy for stencil of sizes (0,4) (off-centered). My questions: I think I am able to answer the question I am just not sure about what is...
  6. Jason Louison

    Engine Finite Heat Release Model (With Heat Transfer) Help

    Hello Physics Forum Users! I have an annoying situation with the Finite Heat Release Equation used to simulate combustion and expansion processes in an internal combustion engine. The equation is as follows: Nomenclature: P = Cylinder Pressure (kPa) θ = Crank Angle (Deg) k = Specific Heat...
  7. C

    Finite element solving of Laplace's equation doesn't converge

    Homework Statement I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
  8. nomadreid

    I For every finite integer sequence there's a pattern- source?

    I have in the back of my head the statement that for every finite sequence of positive integers there exists a pattern (i.e., a generating formula). While this sounds reasonable, I am not sure whether it is true, and if it is true, what the source for this statement is, and how the correct...
  9. mertcan

    Finite element analysis node displacement

    Hi, I would like to ask why center of element displacement is always estimated with polynomial equation involving nodes displacement (like in attachment/picture)? Also I know that if nodes' number increase for an element then displacement of center of element is estimated with higher order...
  10. lfdahl

    MHB Given a,b∈T, a^2−ab+b^2 divides a^2b^2, Prove that T is finite

    Let $T$ be a set of natural numbers such that for any $a, b \in T$, $a^2 − ab + b^2$ divides $a^2b^2$. Prove, that $T$ is finite.
  11. L

    A Convergence order of central finite difference scheme

    For example, when we solve simple 1D Poisson equation by finite difference method, why three point central difference scheme on uniform grid (attached image) is second order method for solution convergence? I understand why approximation of first derivative is second order (and that second...
  12. E

    Finite Element Method: Linear vs. Quadratic Elements & DOF

    Homework Statement Linear and quadratic elements differ because of the extra mid-nodes on quadratic elements. Quadratic elements can "bend", linear can't. How do you define the DOF of a element? (see at solution attempt)Homework Equations - The Attempt at a Solution For example: a linear...
  13. M

    A Is the discretization for the PDE correct?

    Hi PF! I'm trying to finite difference a FTCS PDE $$h_t+\partial_x\left[h^3(h_{xxx}+(Kf'(h)-G)h_x\right] = 0$$ where ##K## and ##G## are constants. ##f'(h) = -(n(h^*/h)^n-m(h^*/h)^m)/h##. Boundary conditions are ##h_x=h_{xxx}=0## at both ends of the ##x## domain (however long you want to make...
  14. M

    A Better way to find Finite Difference

    Given a function ##\psi## having ##N## components, how would you (as fast as possible) construct ##\psi'## also having ##N## components? I thought about taking a forward finite difference approach on the first ##N-1## components of ##\psi## to generate ##\psi'## and then a backward finite...
  15. B

    Nontrivial Finite Rings with No Zero Divisors

    Homework Statement A finite ring with more than one element and no zero divisors is a division ring (Special case: a finite integral domain is a field) Homework EquationsThe Attempt at a Solution Let ##r \in R \setminus \{0\}##, and define ##f : R \setminus \{0\} \to R \setminus \{0\}## by...
  16. RJLiberator

    QM: Potential, Finite Step Function

    Homework Statement A beam of particles of mass m and energy E is incident from the right unto a square well potential given by ##V(x)=-V_0## for ##-a<x<0##, and ##V(x)=0## otherwise. Solve the Schrodinger equation to determine the wave function which describes this situation. Determine the...
  17. L

    How many points of zero probability in a finite well?

    Homework Statement An electron is trapped in a finite potential well that is deep enough to allow the electron to exist in a state with n=4. How many points of (a) zero probability and (b) maximum probability does its matter wave have within the well? Homework Equations For infinite potential...
  18. P

    Electric field above an off-center finite line charge

    Homework Statement There is a finite line charge with length L = 1 meter and linear charge density λ = 1*10^-16 C/m. Point P is h = 70cm above the line charge and distance x from the right end of the line charge. The magnitude and direction of the electric field at point P must be found. The...
  19. B

    Nonmeasurable Set with Finite Outer Measure

    Homework Statement Let ##E## be a nonmeasurable set of finite outer measure. Show that there is a ##G_\delta## set ##G## that contains ##E## for which ##m^*(E)=m^*(G)##, while ##m^*(G-E) > 0##. Homework Equations ##E## is a measurable set if and only if there is a ##G_\delta## set ##G##...
  20. mertcan

    Studying Finite element analysis for Electromagnetism

    HI, initially I would like to put into words that there are sufficient resources, books or lecture videos on YouTube related to finite element analysis especially for structural dynamics (for instance JURGEN BATHE in MIT). But I would like to make you sure that there NO lecture videos or other...
  21. A

    I Controversy about the nature of finite angular displacement

    Hello all, I am having hard time to know if the finite angular displacement really a scalar quantity? In some books they say angular displacement when finite is Scalar and when infinitesimal small is Vector, with direction perpendicular to plane of circle government by right hand rule. I...
  22. A

    I Finite universe, quantized linear momentum, and the HUP....

    Hi all, Yet another question: if the universe is finite, then linear momentum should be quantized (I assume in a similar manner to an infinite potential well since there are boundary conditions). My question is, then, if one computes a value for ##\Delta p## (momentum variance), is the variance...
  23. lfdahl

    MHB A finite number of positive integer solutions 1/x+1/y=p/q

    Given a rational number, $\frac{p}{q}$, show that there are only a finite number of positive integer solutions to the equation: $$\frac{1}{x}+\frac{1}{y}=\frac{p}{q}$$
  24. L

    I The Expanding Universe: A Scientist's Perspective on Infinite vs Finite Space

    I have read some of the other posts about this topic but am still left unsatisfied. Could just be me. :cool: Did the universe, one minute after the big bang, consist of a finite volume of spacetime? If so, then is it not logically inconsistent that the universe can possibly be infinite now...
  25. H

    A Finite difference of fourth order partial differential

    What is a finite difference discretization for the fourth-order partial differential terms \frac{\partial u}{\partial x}k\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}k(x,y)\frac{\partial u}{\partial x} and \frac{\partial u}{\partial x}k(x,y) \frac{\partial u}{\partial y}...
  26. W

    Quantum Mechanics: Outside a (in)Finite Well

    Homework Statement Referencing image attached. I'm not sure how the example arrived at ψ ⇒ 0 at x<0 and >L as K ⇒ ∞ in the limiting case of an infinite potential well. Homework EquationsThe Attempt at a Solution I tried simply applying limits to the wavefuction but in the case x<0, the...
  27. B

    Deductive agruments in finite math: wrong answer in book?

    Homework Statement Identify the following as a valid or an invalid argument. p → q q ∧ r -------------- ∴ ~r → ~p Homework Equations N/A The Attempt at a Solution Truth table values: (a) p → q TTFFTTTT (b) q ∧ r TFFFTFFF (c) a ∧ b TFFFTFFF (d) ~r → ~p TFTFTTTT (e) c → d...
  28. J

    I Finite and infinitesimal Rotations

    Hi, I'm not sure about where I should post this question, so sorry in advance if I posted it in the wrong place. My question is basically this screenshot. So I really have some difficulty in understanding the two equations. I mean how can it not be equal? I understand that rotations are...
  29. Jason Louison

    Automotive Help with finite heat release analysis

    https://www.engr.colostate.edu/~allan/thermo/page8/page8.html The link above takes you to a site I have found to be very helpful in my studies of the Otto cycle, but on this particular page, it depicts an equation for the rate of change of cylinder Pressure vs. crank angle, and in this...
  30. Shakattack12

    Find Year for Pop of 60.4M with Finite Diff Interp

    Homework Statement The problem involves a population in a country: year 1930 1940 1950 1960 1970 Pop 1.0 1.2 1.6 2.8 5.4 (millions) Part A involved finding the population in 1920 using Newton Divided Differences Interpolation (SOLVED) Part B requires finding...
  31. maistral

    Discussions about Finite Element Analysis (PDFs)

    Hi. Can someone point me to a good (and really basic) PDF file or text regarding Finite Element analysis? I would prefer it having an example where it would solve the 1D heat equation or the Laplace equation so I can extend what I know from finite difference methods to it. Sorry if this seems...
  32. Math Amateur

    MHB Finite Extensions - A&F Example 44.2 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 44: Finite Extensions and Constructibility Revisited ... ... I need some help in fully understanding Example 44.2 ... ...Example 44.2 reads as follows...
  33. Math Amateur

    I Finite Extensions - A&F Example 44.2 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 44: Finite Extensions and Constructibility Revisited ... ... I need some help in fully understanding Example 44.2 ... ...Example 44.2 reads as follows: I am trying to fully understand EXACTLY...
  34. M

    Magnetic field of a finite wire increasing with distance?

    I am computing magnetic field around a thick conductor to do railgun force modeling. I am currently re-examining my magnetic field computation, and I have found some confusing results stemming from a fairly simple use of the Biot-Savart Law. The main issue is that the more nuanced application of...
  35. D

    Complete the table for the finite field

    Homework Statement Let ##({a, b, c}, *,+)## be a finite field. Complete the field table for the operations ##*## and ##+## ##\begin{array}{|c|c|c|c|} \hline * & a & b & c \\ \hline a & ? & ? & ? \\ \hline b & ? & ? & ? \\ \hline c & ? & ? & b \\ \hline \end{array}## ##\begin{array}{|c|c|c|c|}...
  36. F

    Electric Field Produced by a Finite Charged Wire

    Homework Statement A charged wire of negligible thickness has length 2L units and has a linear charge density λ. Consider the electric field E⃗ at the point P, a distance d above the midpoint of the wire. What is the magnitude E of the electric field at point P? Throughout this part, express...
  37. Math Amateur

    MHB Existence and Uniqeness of Finite Fields .... Example from D&F ....

    I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...
  38. Math Amateur

    I Existence and Uniqeness of Finite Fields ....

    I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...
  39. Y

    MHB Partial Order Relation where the Set is not Necessarily Finite

    Hello all, I have another question about partial order relations, again, a few statements which are either true or false. R is a partial order relation on a set A which is not necessarily finite. 1) With this order, A has at least one maximal and one minimal elements. 2) If with this order...
  40. F

    Proving the Evenness of Elements Not Equal to Their Own Inverse in Finite Groups

    Homework Statement Prove in any finite group G, the number of elements not equal to their own inverse is an even number. Homework Equations if ab = ba = e, then a = b-1 and b = a-1 The Attempt at a Solution Let S, A, B, be subsets of G where S = A + B. Let a ∈ A s.t. there exists a unique b...
  41. F

    Understanding Finite Element Method: Diagonal Values and Calculating K Matrix

    Homework Statement In the third picture , I don't understand the circled part , add up the values in the diagonal .. How to do that ? I don't understand how to get k13 , k14 , k21 , k22 , k33, k41 and k42 . Homework EquationsThe Attempt at a Solution As we see in the second picture , the k21...
  42. Math Amateur

    MHB Confirm/Critique Solution to Dummit and Foote, Section 13.2: Exercise 2

    I am unsure of my approach to Exercise 2 Dummit and Foote, Section 13.2 : Algebraic Extensions .. I am therefore posting my solution to the part of the exercise dealing with the polynomial g(x) = x^2 + x + 1 and the field F = \mathbb{F}_2 ... ... Can someone please confirm my solution is...
  43. Math Amateur

    Finite Field of Characterisitic p .... ....

    Homework Statement I need help with Exercise 1 of Dummit and Foote, Section 13.2 : Algebraic Extensions .. I have been unable to make a meaningful start on the problem ... ... Exercise 1 of Dummit and Foote, Section 13.2 reads as follows: Homework Equations A relevant definition is the...
  44. Math Amateur

    MHB Help with Dummit & Foote Exercise 1, Section 13.2 - Algebraic Extensions

    I need help with Exercise 1 of Dummit and Foote, Section 13.2 : Algebraic Extensions .. I have been unable to make a meaningful start on the problem ... ... Exercise 1 of Dummit and Foote, Section 13.2 reads as follows: I have been unable to make a meaningful start on this problem ...BUT ...
  45. Wendel

    Path-connectedness for finite topological spaces

    Homework Statement I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? Homework Equations how can I...
  46. maistral

    A Finite difference formulation ideas (journal verification)

    Hi. I am trying to simulate this paper since apparently I have a lot of time. Scrolling down to the last page, he simulated a transient 2D heat conduction plate with composite slabs on it. Darkest one is copper, lighter one is steel, lightest one is glass. If you look closely, the authors said...
  47. maistral

    A 2D Finite Difference formulation in polar coordinates.

    So I have this PDE: d2T/dr2 + 1/r dT/dr + d2T/dθ2 = 0. How do I implement dT/dr || [r = 0] = 0? Also, what should I do about 1/r? This is actually the first time I am going to attack FDF in polar/cylindrical coordinates. I can finite-difference the base equation fairly decently; I am just...
  48. R

    Quantum Mechanics - Question about the Finite Square Well

    Hi, I'm preparing for an exam, and I'm going over past papers. I've solved parts a & b of this question without any problems, however I'm finding it hard to understand part c. I thought of shifting the boundary conditions so I'd have 0 and L in the place of ± L/2, but that would not work...
  49. ShayanJ

    A Finite series and product of Gammas

    In reviewing some calculations, I've arrived at the series: ##S(d)=-\frac 1 {d-1}+\frac 1 2 \frac{d-2}{d-3}-\frac 1 8 \frac{(d-2)(d-4)}{d-5}+\frac 1 {48} \frac{(d-2)(d-4)(d-6)}{d-7}+\dots ## Its an infinite series but because I'm interested in its values for even ##d##s, its actually a finite...
  50. A

    A 2D Finite Difference Convergence Rate Issues

    I have completed a 2D finite difference code in MATLAB that has a domain of (0,1)x(0,1) and has Dirichlet Boundary Conditions of value zero along the boundary. I get convergence rates of 2 for second order and 4 for fourth order. My issue now is that I'm now wanting to change the domain to a...
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