Time step constraints in finite element analysis of the poisson type equation

Time‐step constraints in transient coupled finite element analysis

“…Problems, where the pore water and/or heat flow are multidirectional (i.e. 2D or 3D), were found, by the authors, to be more complex and the critical time-step could not be obtained analytically but had to be determined by trial and error (as also noted by [6]), and therefore are not included in this paper. Even though 1D solutions are less applicable to practical scenarios, the authors found that a thorough understanding of the 1D problems is necessary for extending the theory to more dimensions.…”

“…When linear elements are adopted in a FE analysis of the above heat conduction-convection problem, Equation ( 2 ) can be discretized using the Galerkin method, resulting in: Substituting the shape functions of linear elements into Equation ( 5 ) and then evaluating the integrals, yields: ( 6 ) with the boundary conditions as T1 = Tb and ∂Tn+1/∂x = 0 for any t > 0, and the initial conditions as Ti = T0 at t = 0 for any 1 ≤ i ≤ n+1. For linear elements, the matrix [Cl], which represents the heat content of the material, and the matrices [Kl] and [Dl], which represent the heat transfer due to conduction and convection respectively, can be assembled from the elemental matrices resulting in: As spatial oscillations are of higher importance at the beginning of the analysis, the nodal temperatures Ti after the first time-step, Δt, will be investigated here.…”

“…[1][2][3][4][5][6]) as well as heat conduction problems (e.g. [6][7][8][9][10]) have shown that a lower limit for the size of the time-step exists, below which the solution may exhibit spatial oscillations at the initial stages of the analysis in the regions where the gradient of the solution is steep. These oscillations decay and finally disappear as the gradient of the solution reduces.…”

A three‐point time discretization technique for parabolic partial differential equations

“…They derived the minimum time step for the case of a one-dimensional parabolic partial differential equation based on the discrete maximum principle. Murti et al [5] and Thomas and Zhou [6,7] extended the Vermeer and Verruijt [3] and Rank et al [4] criteria to twodimensional problems adapted to different element types. Later, Yang and Gu [8] derived minimum time step criteria for the Galerkin finite element method applied to one-dimensional parabolic partial differential equations, considering the discretization error of a numerical solution.…”

“…In this case, the Crank-Nicolson method produces oscillations unless t<2/¯ max [2]. There is also a lower bound limit for the time step magnitude below that physically unreasonable results will be obtained using the Crank-Nicolson method [3,5,8].…”