Highly efficient mesh-free approach to simulate the non-linear bending analysis of FG porous beams and sandwich beams with FG face sheets

Radial point interpolation‐Chebychev method with asymptotic numerical method for nonlinear buckling analysis of graphene oxide powder‐reinforced composite beams

“…However, if m is a non-zero value, then m Chebychev polynomials will be added to supplement the interpolation process. [20][21][22][23][24] The RBFs that can be used are: [20][21][22]…”

“…Using the Taylor series development in HOCM, the non-linear discrete Equation ( 21) can be transformed into a sequence of recursive linear equations with the same tangent operator. This allows expressing the unknowns of the problem as a Taylor series expansion truncated at order k with respect to the dimensionless scalar 𝜖 from a known starting solution, 22,23 as follows:…”

“…This could mean, for example, a transition from a stable equilibrium to an unstable one, or the emergence of new oscillatory behavior. 20,22,23,29,[31][32][33][34][35][36][37][38][39][40] Bifurcation analysis has applications in many areas of science and engineering. For example, in physics, bifurcation analysis can be used to understand the behavior of complex systems such as fluid flows.…”

“…However, if $$$ m $$$ is a non‐zero value, then $$$ m $$$ Chebychev polynomials will be added to supplement the interpolation process 20–24 . The RBFs that can be used are: 20–22 $$$$ \left\{\begin{array}{ll}R\left(r,c\right)=\sqrt{r^2+{\left(c{h}_x\right)}^2}& \mathrm{Classical}\ \mathrm{Multiquadrics}\ \left(\mathrm{CMQ}\right)\\ {}R\left(r,c\right)=\sqrt{1+{\left(c{h}_x\right)}^2{r}^2}& \mathrm{Normalized}\ \mathrm{Multiquadrics}\ \left(\mathrm{NMQ}\right)\\ {}R\left(r,k\right)={r}^k\mathit{\ln}(r)& \mathrm{Thin}\ \mathrm{Plate}\ \mathrm{Spline}\ \left(\mathrm{TPS}\right)\\ {}R\left(r,n\right)={e}^{-{(nr)}^2}& \mathrm{Gaussienne}\ \left(\mathrm{EXP}\right)\end{array}\right. $$$$ The characteristic length, denoted as $$$ {h}_x $$$, represents the size of the support domain of a point $$$ x $$$, while $$$ r $$$ indicates the distance between the point $$…”

“…There have been several works that have focused on simulating the properties and behavior of graphene oxide powder (GOP) composite structures. These simulations are often performed using computational methods such as molecular dynamics (MD), 9,10 finite element analysis (FEA), 11,12 meshless methods [13][14][15][16][17] and analytical solution. 18,19 One area of focus in these simulations is on understanding the behavior of the graphene oxide powder within the composite material.…”