Nonlinear impact dynamics and field transfer suitable for parametric design studies

Abstract

PurposeThe purpose of this paper is to provide the methodology for structural design of complex massive structures under impact by a large airplane.Design/methodology/approachUsing case studies, the issues related to multi‐scale modelling of inelastic damage mechanisms for massive structures are discussed, as well as the issues pertaining to the time integration schemes in presence of different scales in time variation of different sub‐problems, brought by a particular nature of loading with a very short duration) and finally the issues related to model reduction seeking to provide an efficient and yet sufficiently reliable basis for parametric studies which are an indispensable part of a design procedure.FindingsSeveral numerical simulations are presented in order to further illustrate the approaches proposed herein. Concluding remarks are stated regarding the current and future research in this domain.Originality/valueProposed design procedure for complex massive engineering structures under impact by a large airplane provides on one side a very reliable representation of inelastic damage mechanisms and external loading represented by the solution of the corresponding contact/impact problem, and on the other side a very efficient basis obtained by model reduction for performing the parametric design studies.

Homogenized Elastic Properties of Graphene for Small Deformations

“…Similarly, the tangent stiffness matrix associated with the angle part of the potential ( i.e. , angle ${\theta }_{jik}$) is defined as:${\mathrm{boldK}}_{i-j-k}=\left[\begin{array}{ccc}{\mathrm{boldP}}_{i,i}^{\theta }& {\mathrm{boldP}}_{i,j}^{\theta }& {\mathrm{boldP}}_{i,k}^{\theta }\\ {\mathrm{boldP}}_{j,i}^{\theta }& {\mathrm{boldP}}_{j,j}^{\theta }& {\mathrm{boldP}}_{j,k}^{\theta }\\ {\mathrm{boldP}}_{k,i}^{\theta }& {\mathrm{boldP}}_{k,j}^{\theta }& {\mathrm{boldP}}_{k,k}^{\theta }\end{array}\right]$ The assembly procedure for the stiffness matrix is performed again, in order to take into account all contributions from pair and angle bonds:${\mathrm{boldK}}_p={\mathrm{double-struckA}}_{i,j}^{n_p}{\mathrm{boldK}}_{i-j},{\mathrm{boldK}}_{\theta }={\mathrm{double-struckA}}_{i,j,k}^{n_{\theta }}{\mathrm{boldK}}_{i-j-k}$ It has been noted before ([33,34]) that an assembly procedure of this kind can be carried out pretty much in the same manner as the standard FEM assembly (e.g., [30]), thus resulting in the model that fits within the standard computer code architecture.…”

“…In the present model, the BC are imposed atom-wise, such that either the displacement, ${\overline{\mathrm{boldd}}}_i$, or the force, ${\overline{\mathrm{boldf}}}_i$, is prescribed. The chosen notation, $\overline{t}$ and $\overline{u}$, is the same for the equivalent notions in continuum mechanics [30], and it is justified in the average sense.…”

“…Unlike conventional FEM for continuum mechanics (e.g., [ 30 ]), we derive and assemble the stiffness and residual matrices by looping over all atoms. In this manner, we can account for the corresponding pairwise and angular potentials defined for each atom interacting with its neighbors.…”

On structural finite element modeling strategies and their influence on the optimization of final constructability of reinforced concrete structures

“…The homogenization associated with the local-global approach is carried out using a weighted least square regression. This method is directly inspired by (Ibrahimbegovic, et al 2009). Different weighting functions are introduced depending on the nature of the deformation.…”

“…The objective of this section is to propose a new fully automated approach. It takes advantage of the field transfer techniques (see (Ibrahimbegovic , et al, 2009)) and conserves the potential energy to consider a complete equivalent set of generalized forces that can be used with classical reinforcement calculation algorithms at a scale larger than the mesh size.…”