Research Article

Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions

¹ 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: michael.stingl@fau.de

Received: 22 June 2017
Accepted: 28 October 2018

Abstract

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.