Volume 52, Number 6, November-December 2018
|Page(s)||2409 - 2431|
|Published online||08 February 2019|
Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department of Mathematics, Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen, Germany
2 Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department of Mathematics, Mathematical Optimization, Nägelsbachstraße 49b, 91052 Erlangen, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 28 October 2018
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.
Mathematics Subject Classification: 34B45 / 68U20 / 74P05 / 90C30 / 90C90
Key words: Timoshenko beam network / analytic solutions / topology optimization / multi-material optimization
© EDP Sciences, SMAI 2019
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