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This article is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads. A solution scheme based on analytic ansatz-functions known to provide analytic solutions for the deformation and rotation of a single beam with given boundary data is extended to the full network. It is demonstrated that the analytic approach is equivalent to a finite element (FE) method where only one element with a suitably chosen shape function per beam is required. The solution of the FE-type system provides analytic solutions at the nodes, from which the solutions along the beams can be reconstructed. Consequently analytic solutions for the network can be computed by a numerical scheme without applying a spacial discretization. While the assembly of the local stiffness matrices is slightly more expensive compared to an FE model using, e.g., linear ansatz-functions, the complexity of the solution of the FE-system is not. This is particularly interesting for topology and material optimization problems formulated on the network. In order to demonstrate the efficiency of the approach a numerical comparison to the case of linear ansatz-functions is provided followed by a series of studies with topology and multi-material optimization problems on networks.
Kufner, Tobias 1 ; Leugering, Günter 1 ; Semmler, Johannes 1 ; Sting, Michael 1 ; Strohmeyer, Christoph 1
@article{M2AN_2018__52_6_2409_0,
author = {Kufner, Tobias and Leugering, G\"unter and Semmler, Johannes and Sting, Michael and Strohmeyer, Christoph},
title = {Simulation and structural optimization of 3d {Timoshenko} beam networks based on fully analytic network solutions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2409--2431},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018065},
mrnumber = {3909807},
zbl = {1419.34107},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018065/}
}
TY - JOUR AU - Kufner, Tobias AU - Leugering, Günter AU - Semmler, Johannes AU - Sting, Michael AU - Strohmeyer, Christoph TI - Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2409 EP - 2431 VL - 52 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018065/ DO - 10.1051/m2an/2018065 LA - en ID - M2AN_2018__52_6_2409_0 ER -
%0 Journal Article %A Kufner, Tobias %A Leugering, Günter %A Semmler, Johannes %A Sting, Michael %A Strohmeyer, Christoph %T Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2409-2431 %V 52 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018065/ %R 10.1051/m2an/2018065 %G en %F M2AN_2018__52_6_2409_0
Kufner, Tobias; Leugering, Günter; Semmler, Johannes; Sting, Michael; Strohmeyer, Christoph. Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2409-2431. doi: 10.1051/m2an/2018065
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