Introduction to the finite element method

By G. Nikishkov. Finite Element Analysis, 2021

2,876 citations recorded by the source corpus.

1. Let Ω = (−1, 1)× (0, 1). We consider the initial boundary value problem ut − uxx = 0, in Ω, (1a) u(±1, t) = 0, for 0 ≤ t ≤ 1, (1b) u(x, 0) = sin(nπx), n ∈ N. (1c) a) Give the analytical solution u(x, t) for general n ∈ N. Hint: Use separation of variables, i. e., u(x, t) = u1(x) · u2(t). b) Consider the problem: Find u(x, 1) for initial data u(x, 0) = sin(nπx), n ∈ N. Is this problem well posed with respect to the L-norm in (−1, 1)? c) Replace ut − uxx = 0 in (1a) by ut + uxx = 0 and redo a) and b).

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/5e58a73db001140b10abb9b6746a578185b69a53

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Citations in the corpus

Computational materials: Multi-scale modeling and simulation of nanostructured materials (2005, neutral): “The method uses a variational technique for solving the differential equations wherein the continuous problem described by the differential equation is cast into the equivalent variation form and the solution is found to be a linear combination of approximation functions [42,30].”
Effects of thermal boundary conditions on natural convection flows within a square cavity (2006, methodology): “The continuity equation (7) will be used as a constraint due to mass conservation and this constraint may be used to obtain the pressure distribution (Basak and Ayappa [21]; Reddy [22]).”
Numerical methods for incompressible viscous flow (2002, neutral): “Emphasis is put on showing the close relationship between (seemingly) different and competing solution approaches for incompressible viscous flow.”
Phase-change heat transfer of single/hybrid nanoparticles-enhanced phase-change materials over a heated horizontal cylinder confined in a square cavity (2017, methodology): “The dimensionless pressure in the momentum equations is estimated by the continuity equation and the penalty parameter (v) as described by [40] which can be written as follows: P 1⁄4 v @U @X þ @V @Y ð21Þ”
Inverse problems in elasticity (2005, methodology): “Accordingly, emphasis will be given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method [28, 125] and the boundary element method [23, 40].”
Room temperature elastic moduli and Vickers hardness of hot-pressed LLZO cubic garnet (2012, methodology): “For example, both Young’s modulus and Poisson’s ratio are often used to construct stiffness matrices for stress–strain calculations via finite element analyses [5].”
Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach (2012, methodology): “One of the advantages of the present approach is to increase easily the order of basic functions without any difficulty whereas the classical FEM requires redistribution of meshing basis.”
Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods (2011, positive): “Using the shape function matrix [ ] N , the displacement is interpolated by [20]:”
Static response and free vibration analysis of FGM plates using higher order shear deformation theory (2010, methodology): “The shape functions which are used in the finite element approximation are mentioned below [28] and the node numbering are shown in Fig.”
Structural Damage Detection Using Modal Curvature and Fuzzy Logic (2009, methodology): “The beam equation is solved for natural frequencies using the FE method [24].”
On the Control of Two-link Flexible Robot Arm with Nonuniform Cross Section (2010, methodology): “Rokui and Khorasani (1994). Singular perturbation theory has also been used for position and force control in Siciliano et al. (2001). Strain feedback and active vibration control, see e.”
Nonparametric estimation of diffusions: a differential equations approach (2012, methodology): “To bound the interpolation error, one can analyse the error element by element, and use a scaling argument and the Bramble-Hilbert lemma (see e.g. p. 277 of Reddy (1984)) to obtain the following bound on the finite element error: Theorem 3.”
The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion (2007, methodology): “Textbooks and monographs on the finite-element method include Strang and Fix (1973), Akin (1986), Strang and Fix (1988), Zienkiewicz and Taylor (1989), Ottosen and Petersson (1992), Hughes (2000), Belytschko et al. (2000), and Reddy (2006).”